\documentclass[11pt]{article} \usepackage{amssymb,amssymb,amscd} \begin{document} % \newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \def\endpf{\hfill$\bullet$\medskip} \newtheorem{rem}[thm]{Remark} \newtheorem{exam}[thm]{Example} \newtheorem{defn}[thm]{Definition} % \newcounter{jolist} \newenvironment{joliste}{\begin{list}{{\rm (\roman{jolist})}}{% \usecounter{jolist}\setlength{\labelwidth}{10mm} \setlength{\leftmargin}{10mm}\setlength{\itemsep}{-4pt} \setlength{\topsep}{0pt}}}{\end{list}} % \newcommand{\al}{\alpha} \newcommand{\avee}{\alpha^\vee} \newcommand{\bvee}{\beta^\vee} \newcommand{\wt}{\widetilde} \newcommand{\delp}{\partial^+} \newcommand{\delm}{\partial^-} \newcommand{\lam}{{\lambda}} \newcommand{\Lam}{{\Lambda}} % \newcommand{\co}{\mathcal{O}} \newcommand{\ck}{\mathcal{K}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cl}{\mathcal{L}} \newcommand{\ci}{\mathcal{I}} \newcommand{\cf}{\mathcal{F}} \newcommand{\cp}{mathcal{P}} % \newcommand{\charc}{\mathop{\rm Char}} \newcommand{\Ker}{\mathop{\rm Ker}} \newcommand{\Lie}{\mathop{\rm Lie}} \newcommand{\Pic}{\rm Pic} \newcommand{\Ch}{\rm Ch} \newcommand{\res}{\mathop{\rm res}} \newcommand{\proj}{\mathop{\rm proj}} \newcommand{\grad}{\mathop{\rm grad}} % \newcommand{\bp}{{\mathbb P}} \newcommand{\bn}{{\mathbb N}} \newcommand{\bz}{{\mathbb Z}} \newcommand{\bq}{{\mathbb Q}} \newcommand{\bb}{{\mathbb B}} \newcommand{\bd}{{\mathbb D}} % \newcommand{\usigma}{{\underline{\sigma}}} \newcommand{\udelta}{{\underline{\delta}}} \newcommand{\ukappa}{{\underline{\kappa}}} \newcommand{\ua}{\underline{a}} \newcommand{\ub}{\underline{b}} \newcommand{\uc}{\underline{c}} \newcommand{\ui}{{\underline{i}}} \newcommand{\uj}{{\underline{j}}} \newcommand{\uka}{{\underline{k}}} \newcommand{\ut}{{\underline{\tau}}} \newcommand{\uk}{{\underline{\kappa}}} \newcommand{\upi}{{\underline{\pi}}} \newcommand{\ueta}{{\underline{\eta}}} \newcommand{\umu}{{\underline{\mu}}} \newcommand{\unu}{{\underline{\nu}}} \newcommand{\us}{{\underline{\sigma}}} \newcommand{\tbar}{{\overline{\tau}}} \newcommand{\kbar}{{\overline{\kappa}}} \newcommand{\sbar}{{\overline{\sigma}}} % \newcommand{\ts}{\tilde{\sigma}} \newcommand{\cR}{{\hat{R}}} % \newcommand{\km}{{\mathfrak{g}}} \newcommand{\lib}{{\mathfrak{b}}} \newcommand{\liu}{{\mathfrak{n}}} \newcommand{\libmin}{{\mathfrak{b}^-}} \newcommand{\liumin}{{\mathfrak{n}^-}} % \title{Richardson Varieties and\\ Equivariant $K$-Theory} % \author{V. Lakshmibai\footnote{V.L. supported by NSF Grant-DMS-9971295.} and P. Littelmann\footnote{P.L. supported by TMR-Grant ERB FMRX-CT97-0100.}} % \maketitle % \begin{abstract} We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of the standard monomial theory as constructed in \cite{Li$_2$}. In fact, the construction given here is very close to the ideas in \cite{$G/P$-V}. Our methods show that in order to develop a SMT for a certain class of subvarieties in $G/B$ (which includes $G/B$), it suffices to have the following three ingredients, a basis for $H^0(G/B,\cl_\lam)$, compatibility of such a basis with the varieties in the class, certain quadratic relations in the monomials in the basis elements. An important tool (as in \cite{$G/P$-V}) will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above. This provides a direct connection to the equivariant $K$-theory (products of classes of structure sheaves with classes of line bundles), where the combinatorially defined notion of standardness gets a geometric interpretation. \end{abstract} %\setcounter{tocdepth}{1}\tableofcontents \section*{Introduction}\label{Introduction} In many respects, the simplest case of the construction of a Standard Monomial basis is the case of the Grassmannian $X$ of linear subspaces of dimension $r$ in $k^n$, embedded into the projective space $\bp(\wedge^r k^n)$ via the Pl\"ucker embedding; let $L$ be the corresponding very ample line bundle on $X$. Then the homogeneous coordinate ring of the embedded variety coincides with the ring $R=\bigoplus_{m=0}^{\infty} H^0(X,L^{\otimes m})$, and this ring admits a nice basis, defined as follows. Let $\{e_1,\ldots,e_n\}$ be the standard basis of $k^n$, then the wedge products $e_\ui=e_{i_1} \wedge \cdots \wedge e_{i_r}$, where $\ui$ varies over the set $I$ of all $r$-tuples such that $1\le i_1<\ldots0$ methods (see Theorem~\ref{vanishandbasis}). Another step in this direction has been proved recently by M. Brion and the first author (cf. \cite{brionlak}). They provide a geometric construction of a (SMT) basis for $H^0(G/P,\cl_\lam)$, compatible with unions of Richardson varieties. The main tool introduced in \cite{brionlak} is a flat $T$-invariant degeneration of the diagonal in $X\times X$ ($X$ being $G/P$) to the union of all products $X_\tau\times X^\tau$, where the $X_\tau$ are the Schubert varieties and the $X^\tau$ are the corresponding opposite Schubert varieties; we also obtain this degeneration as a consequence of the standard monomial basis, see section~\ref{SMTequivariantKtheory}. It is interesting to note that while in \cite{brionlak}, the above degeneration is the starting point towards developing a SMT for Richardson varieties, in this paper such a degeneration is obtained as a consequence of SMT ! In \cite{brionlak}, for any Richardson variety $X_\tau^\kappa$, the authors first construct a basis of the space of sections of $\cl_\lam$ on $X_\tau^\kappa$ vanishing on both of the boundaries $\partial^{\pm}X_\tau^\kappa$ (here, $\partial^{-}X_\tau^\kappa$ is the subvariety of $X_\tau^\kappa$ consisting of the union of all Richardson varieties $X_{\tau'}^{\kappa'}$ such that $\kappa<\kappa'\le\tau'\le\tau$). This basis is then lifted to global sections of $\cl_\lam$ that vanish on all Richardson varieties not containing $X_\tau^\kappa$, which in turn give raise to a basis for $H^0(G/P,\cl_\lam)$; this basis for $H^0(G/P,\cl_\lam)$ is then shown to have compatibility with unions of Richardson varieties. In this article, we show the compatibility of the basis of \cite{Li$_2$} with unions of Richardson varieties (cf. Theorem \ref{vanishandbasis})), and the vanishing of a typical basis vector on $\partial^{\pm}X_\tau^\kappa$ for suitable $\kappa$ and $\tau$. Thus the basis as constructed in \cite{Li$_2$} may be considered as a specially nice basis of the type constructed in \cite{brionlak}. In spite of this relationship between the bases of \cite{brionlak} and \cite{Li$_2$}, there are still some missing links; to be very precise, let $\pi=(\ut,\ua)$ be an L-S path of shape $\lam$, where $\ut=(\tau_0,\ldots,\tau_r)$ is a strictly decreasing sequence in $W/W_P$. The first and last element describe the smallest Richardson variety $X_{\tau_0}^{\tau_r}$ such that the restriction of the path vector $p_\pi$ does not vanish identically. The other terms come up naturally in the quantum group construction and have a suggestive algebraic geometric interpretation in \cite{Li$_2$}, but they do not have yet an interpretation in the construction in \cite{brionlak}. The paper is organized as follows: In the first three sections, we recall the combinatorics of the path model and the construction of the path vector basis: the indexing set $I$ in the Grassmannian case will be replaced by the set of L-S paths $B(\lam)$ of shape $\lam$ (section~\ref{pathmodel}), the basis $\{p_\ui, \ui \in I\}$ will be replaced by the basis $\bb(\lam)$ consisting of path vectors $\{p_\pi, \pi\in B(\lam)\}$ (section~\ref{pathvector}), and the Pl\"ucker relations will be replaced by the quadratic straightening relations (section~\ref{smtsection}). In section~\ref{gradedringsandmodules} we recall the connection between coherent sheaves and graded finitely generated modules over the homogeneous coordinate ring $k[G/P]$ given by the embedding. Using the properties {\it (a)--(c)}, we show in section~\ref{ideal} that the ideal $I_\tau^\kappa$ has a basis given by standard monomials. This is used in section~\ref{idealsheaves} to prove the Pieri--Chevalley type formula for the ideal sheaf $\delp\ci_\tau^\kappa$, and, as a consequence, we obtain in section~\ref{smtandvan} the construction of SMT and the vanishing of higher cohomology groups. In sections~\ref{nonregular} and \ref{SMTnonreg} we discuss the case where the bundle $\cl_\lam$ is only base point free but not necessarily ample. In the last section~\ref{SMTequivariantKtheory}, the relationship between K-theory and SMT is brought out, especially, the computation of the coefficients of the classes of structure sheaves appearing in the product of the class of the structure sheaf of a Schubert variety with the class of a line bundle. The first author is grateful to the Universit\"at Wuppertal for the hospitality extended during her visit in May 2001; it was there that this work originated. \section{Notation}\label{Notation} In this section we fix some standard notation which will be used throughout the paper. The ground field $k$ is supposed to be algebraically closed of arbitrary characteristic. The group $G$ is a semisimple, simply connected algebraic group defined over $k$. We fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. We denote by $B^-$ the opposite Borel subgroup, i.e., $B^-$ is the unique Borel subgroup of $G$ such that $B\cap B^-=T$. The unipotent radical of $B$ is denoted by $U$, and the unipotent radical of $B^-$ is denoted by $U^-$. We use the same notation (but with gothic letters) for the corresponding Lie algebras, i.e., $\km$ is the Lie algebra of $G$, $\lib$ the Lie algebra of $B$, $\libmin$ is the Lie algebra of $\Lie B^-$ and $\liu^\pm$ denotes the Lie algebra of $U^\pm$. The corresponding enveloping algebras are denoted by $U(\km),U(\lib^\pm)$ and $U(\liu^\pm)$. Let $Q\supset B$ be a parabolic subgroup of $G$ containing $B$. The projective variety $G/Q$ admits a finite number of $B$-orbits (as well as a finite number of $B^-$-orbits), they are indexed by the $T$-fixed points in $G/Q$. Let $W$ be the Weyl group of $G$ and let $W_Q$ be the Weyl group of $Q$. The group $W_Q$ can be canonically identified with the subgroup of $W$ generated by the simple reflections $s_\al$ such that $-\al$ is a root of the root system of $Q$. For $\tau\in W/W_Q$ let $e_\tau\in G/Q$ be the corresponding $T$-fixed point. The closure of the $B$-orbit: $X_\tau=\overline{B e_\tau}\subset G/Q$ is the {\it Schubert variety} corresponding to $\tau$, and the closure of the $B^-$-orbit: $X^\tau=\overline{B^- e_\tau}\subset G/Q$, the {\it opposite Schubert variety} corresponding to $\tau$. Let $\tau,\sigma\in W/W_Q$. The {\it Richardson variety} corresponding to the pair $(\tau,\sigma)$ is the (set theoretic) intersection $X_\tau^\sigma = X_\tau\cap X^\sigma$ (with the induced reduced structure). Note taht if $\overline{w}_0\in W/W_Q$ is the class of the longest element $w_0$ in $W$, then $X_{\overline{w}_0}^\sigma=X^\sigma$. Similarly, one has $X_\tau^{id}=X_\tau$. If $\uk=(\kappa_1,\ldots,\kappa_s)$ and $\us=(\sigma_1,\ldots,\sigma_s)$ are sequences of elements in $W/W_Q$, then we denote by $X_{\uk}^{\us}$ the union $X_{\kappa_1}^{\sigma_1} \cup\ldots\cup X_{\kappa_s}^{\sigma_s}$ of Richardson varieties with the induced reduced structure. Let $\Lambda$ be the character group of $T$. We denote by $\Lambda^+$ the dominant weights and by $\Lambda^{++}$ the regular dominant weights. For the parabolic subgroup $Q$, let $\Lambda_Q$ be the subgroup of weights which can be (trivially) extended to characters of $Q$, let $\Lambda^{+}_Q\subset \Lambda_Q$ be the dominant ones and denote by $\Lambda^{++}_Q$ be the $Q$-regular dominant weights, i.e., these dominant weights can not be extended to characters of a parabolic subgroup containing $Q$ properly. For $\lam\in \Lambda$, let $\cl_\lam=G\times_B k_{-\lam}$ be the line bundle on $G/B$ associated to the $B$-character $(-\lam)$. A geometric way to characterize $\Lambda_Q\subset \Lambda$ is to say that $\lam$ is an element of $\Lambda_Q$ if and only if $\cl_\lam$ ``goes down'' to a line bundle $\cl_\lam=G\times_Q k_{-\lam}$ on $G/Q$, and $\lam\in \Lambda_Q^{+}$ if only if $\cl_\lam$ is base point free on $G/Q$, and $\lam\in \Lambda_Q^{++}$ if only if $\cl_\lam$ is an ample line bundle on $G/Q$. \section{The path model and some partial orders}\label{pathmodel} An important combinatorial tool will be the L-S paths of shape $\lam$, we recall quickly the most important features of the path model. Let $\lam\in \Lam^+$ be a dominant weight. Fix a {\it total order ``$\succeq$''} on $W/W_\lam$ refining the {\it Bruhat order ``$\ge$''}. Let $\ut=(\tau_0\succ\ldots\succ\tau_r)$ be a strictly decreasing sequence of elements of $W/W_\lam$ and let $\ua=(0s_{\beta_1}\tau_{i-1}>s_{\beta_2}s_{\beta_1}\tau_{i-1}>\ldots >s_{\beta_{s_i}}\cdots s_{\beta_1}\tau_{i-1}=\tau_i, \] and $a_i\langle\tau_{i-1}(\lam),s_{\beta_1}\cdots s_{\beta_{j-1}}(\beta^\vee_j)\rangle\in\bz$ for all $j=1,\ldots,s_i$. Here $\beta^\vee$ denotes the coroot of $\beta$, and $\langle\mu,\beta^\vee\rangle$ stands for the evaluation of $\mu\in\Lambda$ on the coroot. \end{itemize} For more details on the combinatorics of L-S paths we refer the reader to \cite{Li$_1$}. Let $B(\lam)$ be the set of L-S paths of shape $\lam$. The character of the Weyl module $V(\lam)$ of highest weight $\lam$ can be calculated using the L-S paths: \begin{thm}[\cite{Li$_1$}]\label{pathcharacter} $\charc V(\lam)=\sum_{\pi\in B(\lam)} e^{\pi(1)}$. \end{thm} Let $\pi=(\ut,\ua)$ be an L-S path of shape $\lam$, where $\ut=(\tau_0,\ldots,\tau_r)$. We call $i(\pi)=\tau_0$, the {\it initial direction} and $e(\pi)=\tau_r$, the {\it final direction} of the path. \begin{defn}\rm Let $B(\lam)$ be the set of L--S paths of shape $\lam$. We say that $\pi\in B(\lam)$ is {\it standard on a Richardson variety} $X_\tau^\kappa$ if $\tau\ge i(\pi)$ and $e(\pi)\ge \kappa$, and we denote by $B_\tau^\kappa(\lam)$ the set of all L--S paths of shape $\lam$, standard on $X_\tau^\kappa$. If $\tau$ (resp. $\kappa$) is the class of the longest word in $W$ (resp. $id$), then $\tau$ (resp. $\kappa$) will be omitted and we will write just $B^\kappa(\lam)$ ( resp. $B_\tau(\lam)$). \end{defn} A sequence $\upi=(\pi_1,\ldots,\pi_m)$ of L-S paths of shape $\lam$ is called {\it standard} if \begin{eqnarray}\label{standardsequence} e(\pi_1)\ge i(\pi_2)\ge\ldots \ge e(\pi_{m-1})\ge i(\pi_m). \end{eqnarray} The notion of initial and final direction generalizes to standard sequences of length $m$ as follows: We set $i(\upi)=i(\pi_1)$ and $e(\upi)=e(\pi_m)$. The notion of a standard sequence on a Richardson variety generalizes in the obvious way. Let $\upi(1)=\pi_1(1)+\ldots+\pi_m(1)$ be the {\it weight of such a standard sequence}. The theorem above generalizes to standard sequences: \begin{thm}[\cite{Li$_1$}]\label{standardpathseq} $\charc V(m\lam)=\sum e^{\upi(1)}$, where the sum runs over all standard sequences of length $m$ of L-S paths of shape $\lam$. \end{thm} We need several orderings on the set of convex subsets of $W.\lam$. Let $\pi=(\ut,\ua)$ and $\eta=(\uk,\ub)$ (where $\uk=(\kappa_0,\ldots,\kappa_s)$ and $\ub=(b_1,\ldots,b_s)$) be two convex subsets of $W.\lam$. Induced by the Bruhat order on $W/W_\lam$, we have two types of partial orders on such convex subsets. \begin{itemize} \item We say $\pi\ge\eta$ if $\pi$ is greater than $\eta$ in the weighted lexicographic sense: $\tau_0 >\kappa_0$ or $\tau_0=\kappa_0$ and $a_1>b_1$ or $\tau_0=\kappa_0, a_1=b_1$ and $\tau_1>\kappa_1$, etc. \item We say $\pi\ge^r\eta$ if $\pi$ is greater than $\eta$ in the reverse weighted lexicographic sense: $\tau_r >\kappa_s$ or $\tau_r=\kappa_s$ and $1-a_r>1-b_s$ or $\tau_r=\kappa_s, 1-a_r=1-b_s$ and $\tau_{r-1}>\kappa_{s-1}$, etc. \end{itemize} For a total order $\succeq$ on $W/W_\lam$ we define in the same way total orders $\succeq$ and $\succeq^r$ on the set of convex subset of elements of $W/W_\lam$. We extend these (partial) orders in the obvious way to sequences. Let $\upi=(\pi_1,\ldots,\pi_t)$ and $\ueta=(\eta_1,\ldots,\eta_t)$ be two sequences of weighted subsets. \begin{itemize} \item We say $\upi\ge \ueta$ if $\pi_1>\eta_1$ or $\pi_1=\eta_1$ and $\pi_2>\eta_2$ etc. \item We say $\upi\ge^r \ueta$ if $\pi_t>\eta_t$ or $\pi_t=\eta_t$ and $\pi_{t-1}>\eta_{t-1}$ etc. \end{itemize} The total orders $\succeq$ and $\succeq^r$ are extended to sequences in the same way. \section{The path vector basis}\label{pathvector} Suppose $\lam\in \Lambda_Q^{+}$, then the line bundle $\cl_\lam$ is base point free. Let $V(\lam)$ be the Weyl module of highest weight $\lam$. The parabolic subgroup $Q$ stabilizes the line through the highest weight vector $v_\lam$ and we have a corresponding map $$ G/Q \rightarrow \bp(V(\lam)),\quad gQ\mapsto [gv_\lam]. $$ Further, $H^0(G/Q,\cl_\lam)=V(\lam)^*$ is the dual space of $V(\lam)$. Associated to the combinatorial path model given by the L-S paths $B(\lam)$ of shape $\lam$, we have the basis $$ \bb(\lam)=\{p_\pi\mid\pi\in B(\lam)\}\subset H^0(G/Q,\cl_\lam) $$ given by the path vectors $p_\pi$ as constructed in \cite{Li$_2$}. \begin{rem}\rm The reader not acquainted with the construction should think of these sections in the following way: Let $\pi=(\tau_0,\ldots,\tau_r,a_1,\ldots,a_r)$ be an L-S path of shape $\lam$. Fix $\ell\in\bn$ minimal such that $\ell a_i\in \bn$ for all $i$, and fix weight vectors $p_{\tau_i}\in H^0(G/Q,\cl_\lam)$ of weight $-\tau_i(\lam)$ (these weight spaces are one-dimensional). Then $p_\pi$ can be thought of as an algebraic approximation $$ p_\pi\sim {}^{\ell}\sqrt{p_{\tau_0}^{\ell x_1}p_{\tau_1}^{\ell x_2}p_{\tau_3}^{\ell x_3}\ldots p_{\tau_r}^{\ell x_{r+1}}},\quad x_i=a_i-a_{i-1}, 1\le i\le r+1. $$ Note that in the framework of quantum groups at roots of unity the expression above ``makes sense''. Indeed, ${}^{\ell} \sqrt{p_{\tau_0}^{\ell x_1}\ldots}$ is then the Frobenius splitting map at an $\ell$-th root of unity applied to the product, for details see \cite{Li$_2$}. \end{rem} The path vectors are $T$-weight vectors of weight $-\pi(1)$. Further, the partial orders on L-S paths introduced in the section before are closely related to $B$-stable (respectively $B^-$-stable) submodules spanned by path vectors. More precisely, we call a subset $S^+\subset B(\lam)$ of L-S paths of shape $\lam$ {\it positive saturated} if for all $\pi,\pi'\in S^+$ the following holds: $\bullet$ if $\eta\in B(\lam)$ is such that $\pi'>\eta>\pi$, then $\eta\in S^+$. We say that $S^+\subset B(\lam)$ is {\it maximally positive saturated} if for $\pi\in S^+$ and $\eta\in B(\lam)$ the relation $\eta>\pi$ implies $\eta\in S^+$. \vskip 5pt Similarly, we call a subset $S^-\subset B(\lam)$ of L-S paths of shape $\lam$ {\it negative saturated} if for all $\pi,\pi'\in S^-$ the following holds: $\bullet$ if $\eta\in B(\lam)$ is such that $\pi'>^r\eta>^r\pi$, then $\eta\in S^-$. We say that $S^-\subset B(\lam)$ is {\it maximally negative saturated} if for $\pi\in S^-$ and $\eta\in B(\lam)$ the relation $\pi>^r\eta$ implies $\eta\in S^-$. Let $S^+$ be a maximally positive saturated subset and let $S^-$ be a maximally negative saturated subset of $B(\lam)$. The corresponding path vectors span $T$-submodules of $H^0(G/Q,\cl_\lam)$: $$ M(S^+)=\langle p_\pi\vert\pi\in S^+\rangle \quad{\rm\ and\ }\quad M(S^-)=\langle p_\pi\vert\pi\in S^-\rangle $$ \begin{thm}[\cite{LLM},\cite{Li$_2$}]\label{Bmodule} The $T$--submodule $M(S^+)$ is $B$--stable and the $T$--submodule $M(S^-)$ is a $B^-$--stable submodule of $H^0(G/Q,\cl_\lam)$. \end{thm} \begin{cor} Let $S^+\subset B(\lam)$ be a positive saturated subset and let $M(S^+)$ be the $T$-submodule $\langle p_\pi\vert\pi\in S^+\rangle$ of $H^0(G/Q,\cl_\lam)$ spanned by the corresponding path vectors. Then $M(S^+)$ admits a $B$--module structure which is isomorphic to a subquotient of a $B$--stable filtration of $H^0(G/Q,\cl_\lam)$. \end{cor} {\it Proof\/.} Let $S_1^+=\{\eta\in B(\lam)\mid\exists\,\pi\in S^+:\eta\ge\pi\}$ be the ``closure'' of $S^+$ with respect to $>$, i.e., $S^+_1$ consists of all paths which are greater or equal to some element of $S^+$. The set $S_1^+$ is a maximally positive saturated subset and hence $M(S_1^+)$ is a $B$-submodule of $H^0(G/Q,\cl_\lam)$. The set $S_2^+:=S_1^+ - S^+$ is again a maximally positive saturated subset because: suppose $\pi\in S_2^+$ and $\eta\in B(\lam)$ is such that $\eta>\pi$. By the definition of $S_1^+$ and $S_2^+$, there exists an element $\pi_1\in S^+$ such that $\pi>\pi_1$, so $\eta>\pi>\pi_1$ implies $\eta\in S^+_1$. So either $\eta\in S^+_2$ or $\eta\in S^+$. The latter is not possible because $\eta,\pi_1\in S^+$ would imply $\pi\in S^+$, in contradiction to the assumption $\pi\in S^+_2$, which finishes the proof of the claim. Now $M(S_2^+)$ is hence a $B$-submodule, and the quotient $M(S_1^+)/M(S_2^+)$ is a $B$--module which, as $T$-module, is isomorphic to $M(S^+)$.\endpf The corresponding version for negative saturated subsets holds also, the proof is left to the reader. \begin{cor} Let $S^-\subset B(\lam)$ be a negative saturated subset and let $M(S^-)$ be the $T$-submodule $\langle p_\pi\vert\pi\in S^+\rangle$ of $H^0(G/Q,\cl_\lam)$ spanned by the corresponding path vectors. Then $M(S^-)$ admits a $B^-$--module structure which is isomorphic to a subquotient of a $B^-$--stable filtration of $H^0(G/Q,\cl_\lam)$. \end{cor} For $\tau\in W/W_\lam$ let $v_\tau\in V(\lam)$ be a weight vector of weight $\tau(\lam)$, $v_\tau$ is a so-called {\it extremal weight vector}. The $B$-submodule spanned by the orbit $B.v_\tau$ is called the {\it Demazure module} associated to $\tau$ and is denoted $V_\tau(\lam)$. Similarly, the $B^-$ submodule spanned by the orbit $B^-.v_\tau$ is called the {\it opposite Demazure module} associated to $\tau$ and is denoted $V^\tau(\lam)$. \begin{thm}[\cite{LLM},\cite{Li$_2$}]\label{demazurebasis} The path vector basis is compatible with the Demazure submodules, i.e., the restrictions $\{p_\pi\vert_{V_\tau(\lam)}\mid \pi\in B_\tau(\lam)\}$ form a basis of $V_\tau(\lam)^*$ and the restrictions of the other path vectors vanish on the submodule. Similarly, the restrictions $\{p_\pi\vert_{V^\tau(\lam)}\mid \pi\in B^\tau(\lam)\}$ form a basis of $V^\tau(\lam)^*$ and the restrictions of the other path vectors vanish. \end{thm} Let $\bb(\lam)^*=\{u_\pi\in V(\lam)\vert \pi\in B(\lam)\}$ be the basis of $V(\lam)$ {\it dual} to the path vector basis of $H^0(G/Q,\cl_\lam)$. \begin{cor} The vectors $\{u_\pi\vert \pi\in B_\tau(\lam)\}$ form a basis of the Demazure module $V_\tau(\lam)$, the vectors $\{u_\pi\vert \pi\in B^\tau(\lam)\}$ form a basis of the opposite Demazure module $V^\tau(\lam)$, and the vectors $\{u_\pi\vert \pi\in B_\tau^\sigma(\lam)\}$ form a basis of the intersection $V_\tau^\sigma(\lam)=V_\tau(\lam)\cap V^\sigma(\lam)$. \end{cor} {\it Proof of the corollary\/}: Theorem~\ref{demazurebasis} implies that $V_\tau(\lam)$ is the subspace of $V(\lam)$ orthogonal to $\langle p_\pi\vert i(\pi)\not\le\tau\rangle$. Hence $\{u_\pi\vert i(\pi)\le \tau\}\subset V_\tau(\lam)$, and, again by Theorem~\ref{demazurebasis}, $\langle u_\pi\vert i(\pi)\le \tau\rangle= V_\tau(\lam)$. The proof for the opposite Demazure module is similar and is left to the reader. The statement for the intersection is then an immediate consequence of the fact that the basis is compatible with Demazure and opposite Demazure modules. \endpf \section{Some graded rings and modules}\label{gradedringsandmodules} Suppose $\lam\in \Lambda_Q^{++}$, so the line bundle $\cl_\lam$ is very ample on $G/Q$ and we have a corresponding embedding $G/Q\hookrightarrow \bp(V(\lam))$. Consider the two rings $$ R=\bigoplus_{m\ge 0} H^0(G/Q,\cl_{m\lam})\qquad\hbox{\rm and\ }\qquad k[G/Q]=\bigoplus_{m\ge 0} k[G/Q]_m, $$ where $k[G/Q]$ denotes the homogeneous coordinate ring of the embedding $G/Q\hookrightarrow \bp(V(\lam))$ with the usual grading. Since $\lam$ is ample and $G/Q$ is smooth, one knows that $H^0(G/Q,\cl_{m\lam})=k[G/Q]_m$ for $m\gg 0$. (Actually, by standard monomial theory \cite{Li$_2$} or Frobenius splitting \cite{RR}, one knows that they coincide for all $m$, but we do not need this later.) Correspondingly we denote for a union of Richardson varieties $X_\uk^\us$ the associated rings by $R_\uk^\us$ and $k[X_\uk^\us]$, i.e., $$ R_\uk^\us=\bigoplus_{m\ge 0} H^0(X_\uk^\us,\cl_{m\lam}) \quad\hbox{\rm and\ }\quad k[X_\uk^\us]=\bigoplus_{m\ge 0} k[X_\uk^\us]_m. $$ Let $\ci_\uk^\us\subset\co_{G/Q}$ be the ideal sheaf and let $I_\uk^\us\subset k[G/Q]$ be the homogeneous ideal of $X_\uk^\us\subset G/Q\subset \bp(V(\lam))$. We consider also the graded $R$-module, respectively the graded $k[G/Q]$-module: $$ J_\uk^\us=\bigoplus_{m\ge 0} H^0(G/Q,\ci_\uk^\us\otimes\cl_{m\lam}), \quad\hbox{\rm respectively\ }\quad I_\uk^\us=\bigoplus_{m\ge 0} (I_\uk^\us)_m $$ For Schubert varieties and opposite Schubert varieties we still use the notation $X_\tau, I_\tau, X^\tau, I^\tau$ etc. instead of $X_\tau^{id}, I_\tau^{id}, X^\tau_{\overline{w}_0}, I^\tau_{\overline{w}_0}$ etc. Our aim is to show in the next {\it four sections that the rings $R_\uk^\us$ and $k[X_\uk^\us]$ and the modules $J_\uk^\us$ and $I_\uk^\us$ coincide and have a basis by standard monomials}. \section{Standard monomials and quadratic relations}\label{smtsection} Let $\lam\in\Lambda_Q^{++}$, we use the same notation as in section~\ref{gradedringsandmodules}. We analyze the structure of the homogeneous coordinate ring $k[G/Q]$. We view $k[G/Q]$ as the subalgebra of $R$ generated by $H^0(G/Q,\cl_{\lam}) =k[G/Q]_1$. A monomial of path vectors $p_\upi=p_{\pi_1}\cdots p_{\pi_m}$ is called {\it standard} if the sequence $\upi=(\pi_1,\ldots,\pi_m)$ is standard. The monomial $p_\upi$ is called {\it standard on a Richardson variety $X_\tau^\sigma$} if in addition $\tau\ge i(\upi)\ge e(\upi)\ge\sigma$ for the initial and final directions of $\upi$. The monomial is called {\it standard on a union of Richardson varieties $X_{\uk}^{\us}$} if it is standard on at least one of the $X_\tau^\sigma\subset X_{\uk}^{\us}$. The following relations provide an algorithm to express a non-standard monomial as a linear combination of standard monomials. To be more precise, we need to introduce the wedge product of two convex subsets of $W/W_\lam$. Let $\succeq$ be a the fixed total order refining the Bruhat order. Given two convex subsets $\pi=(\ut,\ua)$ and $\pi'=(\us,\ub)$ of shape $\lam$, let $\uk$ be the sequence obtained from $\ut$ and $\us$ $$ \{\kappa_0,\kappa_1,\ldots\}=\{\tau_0,\ldots,\tau_r\}\cup\{\sigma_0,\ldots,\sigma_s\} $$ by writing the elements $\kappa_j$ in strictly decreasing order with respect to $\succ$. Next let $\uc$ be the strictly increasing sequence of rational numbers obtained from $\ua$ and $\ub$ as follows: $c_i$ is half the sum of the weights of the $\tau_j$ and $\sigma_j$ which are smaller or equal to $\kappa_i$. More precisely: set $c_0,a_0,b_0:=0$. If $\kappa_{i-1}=\tau_{j-1}$ and not equal to one of the $\sigma_m$, then $c_i:=c_{i-1}+(a_j-a_{j-1})/2$; if $\kappa_{i-1}=\sigma_{j-1}$, and not equal to one of the $\tau_m$, then $c_i:=c_{i-1}+(b_j-b_{j-1})/2$; if $\kappa_{i-1}=\tau_{j-1}=\sigma_{m-1}$, then $c_i:=c_{i-1}+((a_j-a_{j-1})+(b_m-b_{m-1}))/2$. \begin{defn}\rm The {\it wedge product} $\pi\wedge\pi'$ of two convex subsets of shape $\lam$ is the convex subset $(\uk,\uc)$ of shape $2\lam$. \end{defn} Note that if $\pi,\pi'$ are L-S paths, then $\pi\wedge\pi'$ is in general not an L-S path. Let now $\pi,\pi'$ be L-S paths of shape $\lam$. We say that $\pi$ and $\pi'$ have the {\it same support} if the sequence $\uk$ in $\pi\wedge\pi'$ is strictly decreasing with respect to the Bruhat ordering $>$. One checks easily the following properties: \begin{lem}\label{wedgeproduct} \begin{joliste} \item[{\rm (i)}] The map $(\pi,\pi')\rightarrow \pi\wedge\pi'$ induces a bijection between the set of standard sequences of length two of L-S paths of shape $\lam$ and the set of L-S paths of shape $2\lam$. \item[{\rm (ii)}] If $\pi,\pi'$ have the same support, then $\pi\wedge\pi'$ is an L-S path of shape $2\lam$. \end{joliste} \end{lem} \begin{rem}\label{mwedge}\rm The wedge product of convex sequences generalizes in the obvious way to $m$-tuples: We define $(\uk,\uc)=\pi_1\wedge\ldots\wedge\pi_m$ as follows: The sequence $\uk$ is the union of the Weyl group cosets occurring in the $\pi_j$, rewritten in strictly decreasing order. Every coset $\kappa_i$ in $\uk$ occurs in at least one path; set $w_i^k=$ the weight of the coset $\kappa_i$, if $\kappa_i$ occurs in $\pi_k$, and set $w_i^k=0$ if $\kappa_i$ does not occur in $\pi_k$. The rational number $c_i$ in $\uc$ corresponding to the coset $\kappa_i$ is $$ c_i={1\over m}\sum_{1\le \ell\le i}\ (\sum_{1\le k\le m} w_\ell^k), $$ the sum over all paths of all weights of the cosets smaller or equal to $\kappa_i$, divided by $m$. One checks easily as above: If the $\pi_i$ are L-S paths of shape $\lam$ and have the same support, then $\wedge\upi=\pi_1\wedge\ldots\wedge\pi_m$ is an L-S path of shape $m\lam$. Further, the map $$ \upi=(\pi_1,\ldots,\pi_m)\longrightarrow \wedge\upi=\pi_1\wedge\ldots\wedge\pi_m $$ induces a bijection between the standard sequences of length $m$ of L-S paths of shape $\lam$, and the set of L-S paths of shape $m\lam$. By construction, one has $i(\upi)=i(\wedge\upi)$ and $e(\upi)=e(\wedge\upi)$, and also the weights coincide: $\upi(1)=\wedge\upi(1)$. \end{rem} If $\pi\wedge\pi'$ is an L-S path of shape $2\lam$, then {\rm (i)} implies there exists a standard sequence $(\eta,\eta')$ such that $\eta\wedge\eta'=\pi\wedge\pi'$. In the following we write just $\pi\wedge\pi'=(\eta,\eta')$ to indicate the corresponding standard sequence. We use the notation $(\eta_1,\eta_2)\succeq \pi_1\wedge\pi_2$ if either $\eta_1\wedge\eta_2\succ \pi_1\wedge\pi_2$, or $\eta_1\wedge\eta_2=\pi_1\wedge\pi_2$ and $\eta_1\succeq \pi_1$. The notation $ \pi_1\wedge\pi_2\succeq^r (\eta_1,\eta_2)$ is defined in the same way: either $\pi_1\wedge\pi_2\succeq^r \eta_1\wedge\eta_2$, or $\pi_1\wedge\pi_2 = \eta_1\wedge\eta_2$ and $\pi_2\succeq^r\eta_2$. \begin{thm}[\cite{LLM}]\label{quadraticrelation} If neither $p_{\pi_1}p_{\pi_2}$ nor $p_{\pi_2}p_{\pi_1}$ is standard, then there exist standard monomials $p_{\eta_1}p_{\eta_2}\in H^0(G/Q,\cl_{2\lam})$ such that $$ p_{\pi_1}p_{\pi_2} =\sum a_{\eta_1,\eta_2} p_{\eta_1}p_{\eta_2} $$ where the coefficient $a_{\eta_1,\eta_2}\not=0$ only if $(\eta_1,\eta_2)\succeq \pi_1\wedge\pi_2\succeq^r (\eta_1,\eta_2)$. Further, $\pi_1\wedge\pi_2 = \eta_1\wedge\eta_2$ is possible only if $\pi$ and $\pi'$ have the same support, and then $a_{\eta_1,\eta_2}=1$ for the standard sequence $(\eta_1,\eta_2)=\pi_1\wedge\pi_2$. \end{thm} Sometimes it is more convenient to formulate the relation as follows: \begin{cor}\label{quadraticrelationII} If neither the monomial $p_{\pi_1}p_{\pi_2}$ nor the monomial $p_{\pi_2}p_{\pi_1}$ is standard, then, in the quadratic relation above, the coefficient of $p_{\eta_1}p_{\eta_2}$ is non-zero only if $\eta_1>\pi_1,\pi_2$ and $\pi_1,\pi_2 >^r\eta_2$. \end{cor} {\it Proof.} This is a consequence of the proof of the theorem above in \cite{LLM}. The main point we will use from the proof is that $a_{\eta_1,\eta_2}\not=0$ only if $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ in the partial order on convex subsets. Now on the one hand, the pairs $(\eta_1,\eta_2)$ are standard sequences, so the wedge product $\eta_1\wedge\eta_2$ is actually independent of the chosen ordering. On the other hand, $\pi_1\wedge\pi_2$ depends on the choice of the total order on $W/W_\lam$, so we are free to choose an appropriate total ordering. Let $\pi_1=(\ut,\ua)$, $\pi_2=(\usigma,\ub)$ and $\eta_1=(\ukappa,\uc)$. Suppose first $\tau_0$ and $\sigma_0$ are not comparable in the Bruhat order. We can choose a total order $\succ_1$ such that $\tau_0\succ_1\sigma_0$ and a total order $\succ_2$ such that $\sigma_0\succ_2\tau_0$. Now $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ implies for the first choice of the total order that $\kappa_0\ge\tau_0$, for the second choice we get as a consequence $\kappa_0\ge\sigma_0$. Since $\tau_0$, $\sigma_0$ are not comparable, we must have strict inequality in both cases, in particular $\eta_1> \pi_1,\pi_2$. If $\tau_0=\sigma_0$, then $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ implies by the definition of the wedge product and the partial order that $\eta_1\ge \pi_1,\pi_2$. So without loss of generality we can assume in the following $\tau_0>\sigma_0$. Since $\eta_1\wedge\eta_2> \pi_1\wedge\pi_2$ implies $\kappa_0\ge\tau_0>\sigma_0$, we see that $\eta_1>\pi_2$. It remains to show: $\eta_1>\pi_1$. If there exists a $j\le r$ such that $\sigma_0\ge \tau_j$ but $\sigma_0\not\ge \tau_{j-1}$, then we can choose a total order such that the sequence of Weyl group cosets in $\pi_1\wedge\pi_2$ is of the form $(\ldots,\tau_{j-1},\sigma_0,\tau_{j},\ldots)$, so $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ implies by the partial lexicographic ordering that necessarily $\eta_1>\pi_1$. Otherwise the sequence of Weyl group cosets in $\pi_1\wedge\pi_2$ is of the form $(\ldots,\tau_{r},\sigma_0,\ldots)$, so $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ implies $\eta_1\ge\pi_1$. But note that $\eta_1=\pi_1$ and $\eta_1\wedge\eta_2>\pi_1\wedge\pi_2$ implies $\eta_2\ge\pi_2$ and hence $\tau_r\ge \sigma_0$, which is not possible by the assumption that $p_{\pi_1}p_{\pi_2}$ is not standard. It follows hence also in this case: $\eta_1>\pi_1$. By replacing $\ge$ and $\succ$ by $\ge^r$ and $\succ^r$, the same arguments show that $\eta_2>^r\pi_1,\pi_2$. \endpf These quadratic relations and the combinatorial character formula are already sufficient to prove: \begin{prop}\label{homcoordinate} The homogeneous coordinate ring $k[G/Q]$ has a vector space basis given by the standard monomials. \end{prop} {\it Proof.} Consider the polynomial algebra $S=k[x_\pi\mid\pi\in B(\lam)]$. We can write any monomial in $S$ as an ordered product: $x_{\pi_1}^{n_1}\cdots x_{\pi_t}^{n_t}$, where $\pi_1\succ^r\ldots\succ^r \pi_t$. We define a monomial order $\succ^r$ on the monomials in S as follows: We say $x_{\pi}\succeq^r x_{\pi'}$ if $\pi\succeq^r\pi'$. If $m_1,m_2$ are two ordered monomials, then we say $m_1\succ^r m_2$ if either the degree of $m_1$ is strictly greater than the degree of $m_2$, or, if the degree of the two coincides, then we say $m_1\succ^r m_2$ if $m_1$ comes before $m_2$ in the reverse lexicographic ordering with respect to $\succeq^r$. Let $m=x_{\pi_1}\cdots x_{\pi_s}$ be an ordered monomial. By abuse of notation we call $m$ a standard monomial in $S$ if the sequence $(\pi_1,\ldots,\pi_s)$ is standard. Denote $E\subset S$ the ideal generated by the type of quadratic relations as in Theorem~\ref{quadraticrelation}, i.e., $E$ is the ideal generated by the relations $$ x_{\pi_1}x_{\pi_2} -\sum a_{\eta_1,\eta_2} x_{\eta_1}x_{\eta_2}, $$ where the set of generators runs over all ordered monomials $x_{\pi_1}x_{\pi_2}$ which are not standard, and the coefficients $a_{\eta_1,\eta_2}$ are as in Theorem~\ref{quadraticrelation}. For $f\in S$, let $in(f)$ be the initial term, i.e., the greatest monomial of $f$ with respect to the chosen monomial ordering $\succ^r$. Let $m=x_{\mu_1}\cdots x_{\mu_s}\in S$ be an ordered monomial which is not standard, say $x_{\mu_j}x_{\mu_{j+1}}$ is not standard. By the relations above, we have, $x_{\mu_j}x_{\mu_{j+1}}-\sum a_{\eta_1,\eta_2} x_{\eta_1}x_{\eta_2}$ is an element of $E$, and $\mu_j\wedge\mu_{j+1}\succ^r (\eta_1,\eta_2)$ for all standard monomials with nonzero coefficient $a_{\eta_1,\eta_2}$. One checks easily that this implies $x_{\mu_j}x_{\mu_{j+1}}\succ^r x_{\eta_1}x_{\eta_2}$ for all standard monomials with nonzero coefficient $a_{\eta_1,\eta_2}$. Set $m_1=x_{\mu_1}\cdots x_{\mu_{j-1}}$ and $m_2=x_{\mu_{j+2}}\cdots x_{\mu_s}$, then $m_1x_{\mu_j}x_{\mu_{j+1}}m_2-\sum a_{\eta_1,\eta_2}m_1x_{\eta_1}x_{\eta_2}m_2$ is an element of $E$ with initial term $m$. So any non-standard monomial $m\in S$ occurs as the initial term of an element of $E$. It follows by Macaulay's Theorem (\cite{Eis}, Theorem 15.3), that the images of the standard monomials form a generating set for the vector space $S/E$. The canonical epimorphism $S\rightarrow k[G/Q]$ defined by $x_\pi\mapsto p_\pi$ factors through $S/E$ (Theorem~\ref{quadraticrelation}). Since $k[G/Q]_m=H^0(G/Q,\cl_{m\lam})\simeq V(m\lam)^*$ for $m\gg 0$, it follows by the character formula (Theorem~\ref{standardpathseq}) that the number of standard monomials is equal to the dimension of $k[G/Q]_m$, so they form in fact a basis. It follows that the standard monomials in $(S/E)_m$ are linearly independent for $m\gg 0$. Note that $\pi=(id)$ is an L-S path of shape $\lam$. For all $\ell>0$ and any standard monomial $x_{\pi_1}\cdots x_{\pi_t}$, the monomial $x_{\pi_1}\cdots x_{\pi_t}x_{id}^\ell$ of degree $t+\ell$ is again standard. So any linear dependence relation between standard monomials of low degree can be made into a linear dependence relation between standard monomials of high degree by multiplying them with a power of $x_{id}$. As a consequence we see that the standard monomials are linearly independent for all $m\ge 0$, and the map $S/E\rightarrow k[G/Q]$ is in fact an isomorphism. \endpf \section{Ideals and coordinate rings}\label{ideal} We analyze now the homogeneous coordinate ring and the defining ideal of a union of Richardson varieties. Throughout this section we assume $\lam\in \Lambda_Q^{++}$ and we consider the embedding $$ X_\uk^\us\subset G/Q \hookrightarrow \bp(V(\lam)). $$ \begin{thm}\label{idealtheorem} The ideal $I_\uk^\us\subset k[G/Q]$ has a basis given by the standard monomials which are not standard on the union of Richardson varieties $X_\uk^\us$, and the homogeneous coordinate ring $k[X_\uk^\us]$ has as basis the restrictions of the monomials standard on $X_\uk^\us$. Further, the scheme theoretic intersection of two such unions is reduced and is again a union of Richardson varieties. In particular, a Richardson variety $X_\tau^\kappa$ is non-empty if and only if $\tau\ge \kappa$, and, in this case, $\dim X_\tau^\kappa=\ell(\tau)-\ell(\kappa)$ and it is the scheme theoretic intersection of the Schubert variety $X_\tau$ and the opposite Schubert variety $X^\kappa$. \end{thm} {\it Proof.} We consider first Schubert varieties. By definition, the linear span of the cone $\hat{X}_\tau$ over $X_\tau\subset \bp(V(\lam))$ is the Demazure submodule $V_\tau(\lam)$. If a standard monomial $p_{\pi_1}\cdots p_{\pi_m}$ is not standard on $X_\tau$, then $p_{\pi_1}$ is not standard on $X_\tau$ and hence vanishes on $V_\tau(\lam)$ (Theorem~\ref{demazurebasis}). It follows that the standard monomials which are not standard on $X_\tau$ lie in $I_\tau$. Since $\cl_\lam$ is ample one knows that $V(m\lam)^*\simeq k[G/Q]_{m}$ for $m\gg 0$, and the latter has a basis given by standard monomials of degree $m$. The monomials which are not standard on $X_\tau$ vanish on $X_\tau$ and hence on $V_\tau(m\lam)$, the linear span of the cone $\hat{X}_\tau \hookrightarrow V(m\lam)$. Theorem~\ref{demazurebasis} (applied to the weight $m\lam$) together with Remark~\ref{mwedge} implies that $\dim V_\tau(m\lam)^*$ is equal to the number of standard monomials on $X_\tau$ of length $m$, so the restrictions form a basis. This proves the linear independence for $m\gg 0$. Now one can use the same arguments as in the proof of Proposition~\ref{homcoordinate} to show that the standard monomials of shape $m\lam$ on $X_\tau$ are linearly independent for all $m\ge 0$. The proof for an opposite Schubert variety is the same. Let $I_\tau^\kappa$ be the ideal $I_\tau^\kappa=I_\tau+I^\kappa\subset k[G/Q]$. The vanishing set of $I_\tau^\kappa$ is $X_\tau^\kappa$. The standard monomials not standard on $X_\tau^\kappa$ form a basis for $I_\tau^\kappa$. Since $p_\pi\not\in I_\tau^\kappa$ if and only if $\tau\ge i(\pi)\ge e(\pi)\ge\kappa$, this shows for $\tau\not\ge\kappa$ that $I_\tau^\kappa=k[G/Q]$ and hence $X_\tau^\kappa=\emptyset$. We assume in the following: $\tau\ge \kappa$. It remains to show that $k[G/Q]/I_\tau^\kappa$ has no nilpotent elements, this will follow from the quadratic relations. Let $f=\sum a_\upi p_\upi\in k[G/Q]$ be a linear combination of standard monomials on $X_\tau^\kappa$, all of the same degree. Let $\ueta$ be a standard sequence of L-S path of shape $\lam$ such that $a_\ueta\not=0$, and suppose that $\ueta$ is minimal with respect to ``$\succ$'' with this property. By applying Theorem~\ref{quadraticrelation} repeatedly, one obtains in $k[G/Q]$: $$ f^m=a_\ueta^m p_{{\wedge^m\ueta}}+\sum a_\usigma p_{\usigma}, $$ where $\wedge^m\ueta$ is the unique minimal (with respect to ``$\succ$'') element with non-zero coefficient which occurs in the expression of $f^m$ as a linear combination of standard monomials. By assumption, $\ueta$ is standard on $X_\tau^\kappa$, so $\wedge^m\ueta$ is standard on $X_\tau^\kappa$ too. It follows that $f^m\not=0$ in $k[G/Q]/I_\tau^\kappa$ for all $m\ge 1$, and hence $k[G/Q]/I_\tau^\kappa$ has no nilpotent elements. As an immediate consequence we see that $X_\tau^\kappa$ is the scheme theoretic intersection $X_\tau\cap X^\kappa$. Let $\tau=\sigma_0>\sigma_1>\ldots>\sigma_r=\kappa$, $r=\ell(\tau)-\ell(\kappa)$ be a strictly decreasing sequence of elements in $W/W_Q$. Then $\pi_i=(\sigma_i)$ is an L-S path of shape $\lam$, and the subring $k[p_{\pi_0},\ldots,p_{\pi_r}]\subset k[X_\tau^\kappa]$ is isomorphic to a polynomial ring in $r+1$ variables since all monomials are standard. There are only a finite number of such strictly decreasing sequences, and these are the only generators such that products of powers are again standard. So the number of standard monomials of a fixed degree $m$ is approximately the same as the number of monomials of degree $m$ of a polynomial ring in $r+1$ variables. So the Hilbert polynomial of $k[X_\tau^\sigma]$ is of degree $r$ and hence $\dim X_\tau^\sigma=r$. Next consider a pair of sequences $\uk=(\kappa_1,\ldots,\kappa_t)$ and $\usigma=(\sigma_1,\ldots,\sigma_t)$ such that $\kappa_j\ge\sigma_j$ for all $j$, and let $X_\uk^\us$ be the corresponding union of Richardson varieties. Denote $I_\uk^\us\subset k[G/Q]$ its vanishing ideal, then $p_\upi\in I_\uk^\us$ for all standard monomials $p_\upi$ not standard on $X_\uk^\us$. If $f\in I_\uk^\us$, then write $f=\sum a_\upi p_\upi$ as a linear combination of standard monomials. Since the restriction of a standard monomial to a Richardson variety either vanishes or is standard and forms part of a basis, the vanishing of $f$ on $X_\uk^\us$ implies that $f$ vanishes on all $X_\kappa^\sigma\subset X_\uk^\us$. Hence none of the $p_\upi$ is standard on $X_\uk^\us$, so the standard monomials $p_\upi$, not standard on $X_\uk^\us$, form a basis for $I_\uk^\us$. The combinatorial description of the ideals shows immediately that the scheme theoretic intersection $X_\uk^\us\cap X_{\uk'}^{\us'}$ of two such unions is reduced and is again a union of Richardson varieties. \endpf \begin{rem}\rm The dimension formula and the irreducibility for Richardson varieties also follow more geometrically from the description of $Be_\tau\cap B^-e_\kappa$ by Deodhar \cite{Deo}, who showed that the intersection of the latter is isomorphic to a product of $k$'s and $k^*$'s, and the proof by Richardson \cite{Ri}, that the closure of such an intersection is a Richardson variety; the reducedness of intersections and unions of Richardson varieties follow from \cite{Ram1}. Further, using the methods of \cite{RR,Ram1,Ram2}, one may also deduce that Richardson varieties are normal and Cohen-Macaulay (see also \cite{brionlak}, Lemma 1). As remarked in the Introduction, one of the main aims of this paper is to present standard monomial theoretic proofs for results on Richardson varieties and their unions, proof of Theorem \ref{idealtheorem} being one such example; of course, along the way, standard monomial theoretic approach also gives us additional results, for instance, a standard monomial basis for ideals of Richardson varieties and their unions (as in \ref{idealtheorem}). \end{rem} \section{Filtrations of ideals and ideal sheaves}\label{idealsheaves} The standard monomial bases of the coordinate rings and defining ideals suggest certain vector space decompositions: For example, let $X_\tau^\sigma$ be a Richardson variety and suppose $\pi$ is an L-S path of shape $\lam\in \Lambda_Q^{++}$ standard on $X_\tau^\sigma$. The standard monomials of degree $m$ on $X_\tau^\sigma$ starting with $p_\pi$ are all of the form $p_\pi p_{\upi'}$, where $p_{\upi'}$ is a standard monomial of degree $(m-1)$ such that $$ e(\pi)\ge i(\upi')\ge e(\upi')\ge \sigma. $$ As a vector space, the space of standard monomials of degree $m$ on $X_\tau^\sigma$ starting with $p_\pi$ can be identified with the space of standard monomials of degree $(m-1)$, standard on $X_{e(\pi)}^\sigma$. The aim of this section is to formulate this vector space decomposition more precisely in terms of filtrations and associated graded modules and sheaves. We have two types of boundaries of a Richardson variety $X_\tau^\sigma$, the {\it positive} and the {\it opposite} or {\it negative} boundary: $$ \delp X_\tau^\sigma=\bigcup_{\sigma\le\kappa<\tau} X_\kappa^\sigma\qquad\hbox{\rm and\ }\qquad \delm X_\tau^\sigma=\bigcup_{\sigma<\kappa\le\tau} X_\tau^\kappa $$ Correspondingly, let $\delp I_\tau^\sigma\subset k[X_\tau^\sigma]$ be the defining ideal of $\delp X_\tau^\sigma$ and let $\delm I_\tau^\sigma\subset k[X_\tau^\sigma]$ be the defining ideal of $\delm X_\tau^\sigma$. The standard monomials (or rather the images), standard on $X_\tau^\sigma$, form a basis of $k[X_\tau^\sigma]$. By Theorem~\ref{idealtheorem}, $\delp I_\tau^\sigma$ has a basis given by the standard monomials $p_{\upi}$ on $X_\tau^\sigma$ such that $i(\upi)=\tau$, and $\delm I_\tau^\sigma$ has a basis given by the standard monomials $p_{\upi}$ on $X_\tau^\sigma$ such that $e(\upi)=\sigma$. The partial orders on L-S paths introduced in section~\ref{pathmodel} are not only strongly related to the $B$--module structure (see section~\ref{pathvector}) but also to the ideal structure of the homogeneous coordinate ring of Richardson varieties. %We fix a total order $\succ$ on $W/W_\lam$ refining the Bruhat order, and we denote %by $\succ$ and $\succ^r$ the corresponding total order on convex subsets $(\uk,\ua)$. % Denote by $S_\tau^+$ {\it the set of L-S paths of shape $\lam$ such that $i(\pi)=\tau$}. Note that this set is positive saturated (see section~\ref{pathvector}) with a unique maximal element: the path $(\tau)$. \begin{thm}\label{idealfiltration} Let $X_\tau^\sigma$ be a Richardson variety and let $\delp X_\tau^\sigma$ be its positive boundary. Consider a sequence of subsets $S_j\subset S_\tau^+$ such that $\vert S_j\vert=j$, $(\tau)\in S_j$ for $j>0$, and $S_j$ is positive saturated for all $j$: $$ S_0=\emptyset\subset S_1\subset\ldots \subset S_{N-1}\subset S_N= S_\tau^+. $$ Set $(\delp I_\kappa^\sigma)_j=\sum_{\pi\in S_j} k[X_\kappa^\tau]p_{\pi}$, then the filtration by $T$-stable ideals $$ 0\subset (\delp I_\kappa^\sigma)_1\subset\ldots\subset (\delp I_\kappa^\sigma)_N =\delp I_\kappa^\sigma $$ is such that the subquotients are as $k[X_\kappa^\sigma]$-modules and $T$-modules isomorphic to: $$ (\delp I_\kappa^\sigma)_j/(\delp I_\kappa^\sigma)_{j-1}\simeq k[X_{e(\pi)}^\sigma](-1)\otimes\chi_{-\pi(1)}, \quad \{\pi\}=S_j-S_{j-1}, $$ where $k[X_{e(\pi)}^\sigma]$ denotes the homogeneous coordinate ring of $X_{e(\pi)}^\sigma\subset X_{\kappa}^\sigma$. The isomorphism is induced by the graded morphism $$ (\delp I_\kappa^\sigma)_j\rightarrow k[X_{e(\pi)}^\sigma](-1), \qquad \big(p_\pi f_\pi+\sum_{\eta\in S_{j-1}} p_{\eta} f_\eta\big)\mapsto f_\pi\vert_{X_{e(\pi)}^\sigma}. $$ \end{thm} \begin{rem}\rm \begin{joliste} \item[{\rm(i)}] There is a corresponding obvious version of the theorem for the ideal $\delm I_\kappa^\sigma$. \item[{\rm(ii)}] If $\sigma=id$, i.e., $X_\kappa^\sigma= X_\kappa$ is a Schubert variety, then the filtration can in addition be chosen to be $B$-equivariant. \item[{\rm(iii)}] If $\kappa$ is the class of the longest word in $W/W_\lam$, i.e., $X_\kappa^\sigma= X^\sigma$ is an opposite Schubert variety, then the filtration for the ideal $\delm I_\kappa^\sigma$ can in addition be chosen to be $B^-$-equivariant. \end{joliste} \end{rem} {\it Proof.} Consider the vector space $J_j\subset k[X_\kappa^\sigma]$ spanned by all standard monomials starting with a $p_{\pi}$, $\pi\in S_j$, then $J_j\subset (\delp I_\kappa^\sigma)_j$. Since $S_j$ is positive saturated, Corollary~\ref{quadraticrelationII} implies in fact that $J_j$ is an ideal and hence, by the definition, $J_j=(\delp I_\kappa^\sigma)_j$. So the latter has the standard monomials on $X_\kappa^\sigma$ starting with a $p_{\pi}$, $\pi\in S_j$, as a basis. This shows that the subquotient $(\delp I_\kappa^\sigma)_j/(\delp I_\kappa^\sigma)_{j-1}$ has as basis the images of the standard monomials on $X_\kappa^\sigma$ starting with $p_{\pi}$, $\{\pi\}=S_j-S_{j-1}$. Let $f=\sum_{\eta\in S_j} p_\eta f_\eta$ be an element of $(\delp I_\kappa^\sigma)_j$. Let $\pi\in S_j$ be the unique element such that $\pi\not\in S_{j-1}$. Write $f_\pi=\sum a_\ueta p_\ueta \in k[X_\kappa^\sigma]$ as a linear combination of standard monomials. Let $f_{\pi,1}=\sum a_\ueta p_\ueta$ be the sum of those summands in the expression above such that $p_{\pi}p_\ueta$ is standard, and set $f_{\pi,2}=f_\pi-f_{\pi,1}$. The quadratic relations (Corollary~\ref{quadraticrelationII}) imply that $p_{\pi}f_{\pi,2}\in (\delp I_\kappa^\sigma)_{j-1}$, and Theorem~\ref{idealtheorem} shows that the restriction of $f_\pi$ to $X_{e(\pi)}^\sigma$ coincides with the corresponding restriction of $f_{\pi,1}$. It follows that the map $$ (\delp I_\kappa^\sigma)_j\rightarrow k[X_{e(\pi)}^\sigma](-1), \qquad \big(p_\pi f_\pi+\sum_{\eta\in S_{j-1}} p_{\eta} f_\eta\big)\mapsto f_\pi\vert_{X_{e(\pi)}^\sigma} $$ is a well defined graded $k[X_\kappa^\sigma]$-module homomorphism. One checks easily that the map is surjective, has $(\delp I_\kappa^\sigma)_{j-1}$ as kernel and is $T$-equivariant up to a twist by the character corresponding to the weight of $p_{\pi}$. But this is the same as to say that that the homomorphism induces an isomorphism between the $T$- and $k[X_\kappa^\sigma]$-modules $(\delp I_\kappa^\sigma)_j/(\delp I_\kappa^\sigma)_{j-1}$ and $k[X_{e(\pi)}^\sigma](-1)\otimes\chi_{-\pi(1)}$. \endpf We consider the corresponding sheaves. Since $\cl_\lam$ is a very ample sheaf on $G/Q$ and the variety is smooth, one knows that $H^0(G/Q,\cl_{m\lam}) =k[G/Q]_m$ for $m\gg 0$. Let $\co_{X_\kappa^\sigma}$ be the structure sheaf of $X_\kappa^\sigma\subset G/Q$ and let $\ci_\kappa^\sigma\subset \co_{G/Q}$ be its sheaf of ideals, i.e., we have the exact sequence $$ 0\rightarrow \ci_\kappa^\sigma\rightarrow\co_{G/Q}\rightarrow \co_{X_\kappa^\sigma}\rightarrow 0. $$ Again $\cl_\lam$ being ample, for $m\gg 0$ we get an exact sequence: $$ 0\rightarrow H^0(G/Q,\ci_\kappa^\sigma\otimes\cl_{m\lam})\rightarrow H^0({G/Q},\cl_{m\lam})\rightarrow H^0(X_\kappa^\sigma,\cl_{m\lam})\rightarrow 0 $$ and hence an isomorphism $k[X_\kappa^\sigma]_m\simeq H^0(X_\kappa^\sigma,\cl_{m\lam})$ for $m\gg 0$. Similarly, let $\delp \ci_\kappa^\sigma$ be the sheaf of ideals of $\delp X_\kappa^\sigma\subset X_\kappa^\sigma$. The exact sequence $$ 0\rightarrow \delp \ci_\kappa^\sigma \rightarrow\co_{X_\kappa^\sigma}\rightarrow \co_{\delp X_\kappa^\sigma}\rightarrow 0. $$ induces for $m\gg 0$ an exact sequence: $$ 0\rightarrow H^0(X_\kappa^\sigma, {\delp \ci_\kappa^\sigma}\otimes\cl_{m\lam})\rightarrow H^0({X_\kappa^\sigma},\cl_{m\lam})\rightarrow H^0(\delp X_\kappa^\sigma,\cl_{m\lam})\rightarrow 0 $$ and hence isomorphisms $H^0(\delp X_\kappa^\sigma,\cl_{m\lam})\simeq k[\delp X_\kappa^\sigma]_m$ for $m\gg 0$. For the ideal %%%%%%%%%%%%%%%%%%%%%%%%%%% $\delp I_\kappa^\sigma$ we get isomorphisms $H^0(X_\kappa^\sigma, {\delp\ci_\kappa^\sigma}\otimes\cl_{m\lam})\simeq [\delp I_\kappa^\sigma]_m$ for $m\gg 0$, where the grading is defined by $[\delp I_\kappa^\sigma]_m=\delp I_\kappa^\sigma \cap k[X_\kappa^\sigma]_m$. %%%%%%%%%%%%%%%%%%%%%%%%% Set $R=\bigoplus_{m\ge 0} H^0(G/Q,\cl_{m\lam})$. Since $\cl_\lam$ is ample, coherent sheaves on $G/Q$ correspond to finitely generated graded $R$-modules. If $M$ is such a module, the sheaf is given locally by quotients (in the localization of $M$) of homogeneous elements of $M$ and $R$ of the same degree. %The equivalence class of such a local section does not change if the enumerator and the denominator %of such a quotient are multiplied by the same homogeneous element from $R_\kappa^\sigma$ (which is locally %invertible). So one can %represent such a local section always as a quotient of homogeneous elements of high degree. Recall that graded modules which differ only in a finite number of graded components give the same coherent sheaf. It follows that the filtration of $\delp I_\kappa^\sigma$ (Theorem~\ref{idealfiltration}) provides a filtration of $\delp\ci_\kappa^\sigma$ as $\co_{G/Q}$-module and as a $T$-sheaf: \begin{cor}[Pieri-Chevalley type formula]\label{idealsheaffiltration} The filtration of the vanishing ideal $\delp I_\kappa^\sigma$ induces a filtration of $\delp\ci_\kappa^\sigma$ such that the associated graded sheaf is, as $T$-equivariant sheaf of $\co_{G/Q}$-modules, the direct sum $$ \grad \delp\ci_\kappa^\sigma=\bigoplus \co_{X_{e(\pi)}^\sigma}(-1)\otimes \chi_{-\pi(1)} $$ where the sum runs over all L-S paths $\pi$ of shape $\lam$, standard on $X_\kappa^\sigma$ and such that $i(\pi)=\kappa$. \end{cor} \section{Vanishing theorems and standard monomials}\label{smtandvan} As before, we assume $\lam\in \Lambda_Q^{++}$. \begin{thm}\label{vanishandbasis} Let $X_\uk^\us$ be a union of Richardson varieties. \begin{joliste} \item[{\rm (i)}] $H^i(X_\uk^\us,\cl_{m\lam})=0$ for all $i\ge 1$ and all $m\ge 1$, and for an irreducible variety one has in addition $H^i(X_\kappa^\sigma,\co_{X_\kappa^\sigma})=0$ for all $i\ge 1$. \item[{\rm (ii)}] The standard monomials, standard on $X_\uk^\us$ of degree $m$, form a basis for $H^0(X_\uk^\us,\cl_{m\lam})$ for all $m\ge 1$. \end{joliste} \end{thm} {\it Proof\/} The proof is by induction on the maximal dimension of the irreducible components and on the number of irreducible components of maximal dimension and on $m$. The theorem holds if $\dim X_\uk^\us=0$. Assume that the theorem holds for all unions of Richardson varieties of dimension smaller than $n$. Suppose now $\dim X_\kappa^\sigma=n$. Consider the exact sequence $$ 0\rightarrow \delp \ci_\kappa^\sigma\otimes\cl_\lam \rightarrow\co_{X_\kappa^\sigma}\otimes\cl_\lam \rightarrow \co_{\delp X_\kappa^\sigma}\otimes\cl_\lam\rightarrow 0. $$ Since $\delp X_\kappa^\sigma$ is a union of Richardson varieties of smaller dimension, the vanishing theorem holds and hence $H^i(\delp X_\kappa^\sigma,\cl_\lam)=0$ for $i\ge 1$. Moreover, by induction, the global sections on $\delp X_\kappa^\sigma$ can be lifted to global sections on $G/Q$, so the restriction map $H^0(X_\kappa^\sigma,\cl_\lam) \rightarrow H^0(\delp X_\kappa^\sigma,\cl_\lam) $ is surjective. By Corollary~\ref{idealsheaffiltration}, the sheaf $\delp \ci_\kappa^\sigma\otimes\cl_\lam$ admits a filtration such that the associated graded is just a direct sum of structure sheafs of Richardson varieties of the form $\co_{X_{\kappa'}^\sigma}$, where $\kappa\ge \kappa'$. By induction, for $\kappa'<\kappa$ the vanishing for the higher cohomology groups holds. The associated graded has only one subquotient isomorphic to $\co_{X_{\kappa}^\sigma}$, the contribution coming from the path $(\kappa)$. So the long exact sequence in cohomology splits into isomorphisms $H^i(X_\kappa^\sigma,\co_{X_{\kappa}^\sigma}) \simeq H^i(X_\kappa^\sigma,\cl_\lam)$ for $i\ge 1$ and a short exact sequence $$ 0\rightarrow H^0(X_\kappa^\sigma,\delp \ci_\kappa^\sigma\otimes\cl_\lam) \rightarrow H^0(X_\kappa^\sigma,\cl_\lam) \rightarrow H^0(\delp X_\kappa^\sigma,\cl_\lam)\rightarrow 0, $$ so $\dim H^0(X_\kappa^\sigma,\cl_\lam)=\dim H^0(\delp X_\kappa^\sigma,\cl_\lam)+ \dim H^0(X_\kappa^\sigma,\delp \ci_\kappa^\sigma\otimes\cl_\lam)$. Consider the right hand side of the equation. By induction, the first term is equal to the number of L-S paths of shape $\lam$ standard on $\delp X_\kappa^\sigma$, and the second is by the filtration equal to the number of L-S paths of shape $\lam$ standard on $X_\kappa^\sigma$ and such that $i(\pi)=\kappa$. This together is the number of L-S paths of shape $\lam$ standard on $X_\kappa^\sigma$. Since $k[X_\kappa^\sigma]_1\hookrightarrow H^0(X_\kappa^\sigma,\cl_\lam)$, this proves that the path vectors form a basis of $H^0(X_\kappa^\sigma,\cl_\lam)$. For $m>1$ one gets an exact sequence: $$ 0\rightarrow \delp \ci_\kappa^\sigma\otimes\cl_{m\lam} \rightarrow\co_{X_\kappa^\sigma}\otimes\cl_{m\lam} \rightarrow \co_{\delp X_\kappa^\sigma}\otimes\cl_{m\lam}\rightarrow 0. $$ The same arguments as above (induction on the dimension or on $m$ and the filtration of $\delp \ci_\kappa^\sigma\otimes\cl_{m\lam}$) show that the associated long exact sequence in cohomology splits into isomorphisms $H^i(X_\kappa^\sigma,\cl_{(m-1)\lam}) \simeq H^i(X_\kappa^\sigma,\cl_{m\lam})$ for $i\ge 1$ and a short exact sequence for the global sections. The same counting argument as above proves that the standard monomials form a basis of $H^0(X_\kappa^\sigma,\cl_{m\lam})$. Further, for $i\ge 1$ we have isomorphisms: $$ H^i(X_\kappa^\sigma,\co_{X_{\kappa}^\sigma})\simeq H^i(X_\kappa^\sigma,\cl_{\lam}) \simeq H^i(X_\kappa^\sigma,\cl_{m\lam}),\ \forall m\ge 1 $$ Since $\cl_\lam$ is ample, it follows that $H^i(X_\kappa^\sigma,\cl_{m\lam})=0$ for all $i\ge 1, m\ge 0$. Suppose now $X_\uk^\us$ is a union of Richardson varieties. Let $X_\kappa^\sigma$ be an irreducible component of maximal dimension and denote $X_{\uk'}^{\us'}$ the union of the remaining irreducible components. We have an exact sequence: $$ 0\rightarrow \co_{X_\uk^\us}\rightarrow \co_{X_\kappa^\sigma}\oplus \co_{X_{\uk'}^{\us'}}\rightarrow \co_{X_{\uk'}^{\us'}\cap X_\kappa^\sigma}\rightarrow 0. $$ If we tensor the sequence with $\cl_{m\lam}$, $m\ge 1$, then we know by induction on the dimension, respectively the number of irreducible components of maximal dimension that the higher cohomology groups of the second and the last term vanish, and further, for the global sections the last map is surjective. It follows that the higher cohomology groups $H^i(X_\uk^\us,\cl_{m\lam})$ vanish. It remains to count the dimensions of the spaces of global sections. Since one has standard monomial theory for the last two terms by induction on the dimension, respectively the number of irreducible components of maximal dimension, one checks easily that in fact $\dim H^0(X_\uk^\us,\cl_{m\lam})$ is equal to the number of standard monomials on $X_\uk^\us$ of degree $m$. The same argument as above proves hence that the standard monomials form a basis of $\dim H^0(X_\uk^\us,\cl_{m\lam})$. \endpf \begin{rem}\rm The result (i) in Theorem \ref{vanishandbasis} is also proved in \cite{brionlak} (cf. \cite{brionlak}, Lemma 5). \end{rem} \section{The non-regular case and pointed unions}\label{nonregular} Fix a dominant weight $\lam\in \Lam_Q^+$. Note that in the general case not all results carry over. We have the following two examples (cf. \cite{brionlak}, Remark following Lemma 5). Consider $G=SL_2$, so $G/B=\bp^1$. Let $X_\uk^\us=\{0,\infty\}$ be the union of the two $T$-fixed points. Then $\dim H^0(G/B,\co_{G/B})=1$ but $\dim H^0(X_\uk^\us,\co_{X_\uk^\us})=2$, so the restriction map for global sections is not surjective. In $\bp^1\times\bp^1$, let $X_\uk^\us$ be the union of the four $\bp^1$'s: $\{0\}\times \bp^1$, $\{\infty\}\times \bp^1$, $\bp^1\times \{0\}$ and $\bp^1\times \{\infty\}$, then $H^1(X_\uk^\us,\co_{X_\uk^\us})\not=0$. But we get a SMT (standard monomial theory) and vanishing theorems also in the non-regular case for a special class of unions of Richardson varieties. In the inductive procedure used to construct standard monomial theory, we need a class of varieties which includes {\it i)} all Richardson varieties, {\it ii)} their boundaries $\delp X_\tau^\kappa$ and $\delm X_\tau^\kappa$, and {\it iii)} if $Y$ is in this class and $Y=X\cup Y'$ is such that $X$ is irreducible, then $X$, $Y\cap X$ and $Y'$ are also in this class. The example above shows: we can expect that the global restriction map is surjective for all varieties in this class only if it is not possible to construct a non-trivial union of points by using the operations in {\it ii)} and {\it iii)}. This leads to the following definition: \begin{defn}\label{pointedRichardson} {\it A union of Richardson varieties $Y$ is called pointed\/} if there exists a $\kappa\in W/W_Q$ such that $Y=\bigcup_j X_\kappa^{\sigma_j}$, or if there exists a $\sigma\in W/W_Q$ such that $Y=\bigcup_j X_{\kappa_j}^\sigma$. \end{defn} Simple examples for such pointed unions are arbitrary unions of Schubert varieties or arbitrary unions of opposite Schubert varieties. It is easy to check that the class of pointed unions of Richardson varieties has the three properties above. A first step to construct SMT is the generalization of Corollary~\ref{idealsheaffiltration}, but first the definition of the boundary has to be adjusted. For $\tau\in W/W_Q$, set $\tbar\equiv\tau \bmod W_\lam$, and denote $p_\tbar$ the extremal weight vector in $H^0(G/Q,\cl_\lam)$ corresponding to the L-S path $(\tbar)$ of type $\lam$. \begin{prop}\label{emptylambdaboundary} Let $P_\lam\supset Q$ be the parabolic subgroup associated to the weight $\lam$, and for $\tau\ge \kappa\in W/W_Q$ consider the projection: $$ \begin{array}{rccc} \phi:& X_\tau^\kappa&\longrightarrow &G/P_\lam\\ &\hskip 15pt \searrow & & \nearrow\hskip 15pt\\ & & G/Q & \\ \end{array} $$ Either $X_\tau^\kappa\subset \phi^{-1}(e_\tbar)$, in which case the line bundle $\cl_\lam\vert_{X_\tau^\kappa}$ is trivial, or the preimage of set of zeros of $p_\tbar$ below, called the boundary of $X_\tau^\kappa$ with respect to $\lam$ (or just the $\lambda$-{\it boundary}), is not empty: $$ \delp_\lam X_\tau^\kappa = \{y\in X_\tau^\kappa\mid p_\tbar(y)=0\}\not=\emptyset $$ \end{prop} \begin{rem}\rm We use the notation $ \delp_\lam X_\tau^\kappa$ in the following for the corresponding variety with its induced reduced structure. \end{rem} {\it Proof.\/} Let $Z=\phi(X_\tau^\kappa)$ be the image of the $T$-equivariant map $\phi$. If $Z=e_\tbar$, then $X_\tau^\kappa\subset \phi^{-1}(e_\tbar)$ and the restriction of $\cl_\lam$ to the fibre is constant. Otherwise $\dim Z\ge 1$ and hence $Z$ admits at least two $T$-fixed points, say $e_\tbar$ and $e_\sbar$ for some $\sbar\not=\tbar$. Let $y\in X_\tau^\kappa$ be such that $\phi(y)=e_\sbar$, then $p_\tbar(y)=p_\tbar(e_\sbar)=0$ and hence $y\in \delp_\lam X_\tau^\kappa$, so the latter is not empty. \endpf \begin{rem}\label{boundarydescription}\rm Let $\tbar$ be as above and let $\delp X_\tbar=\bigcup X_{\sbar_i}\subset G/P_\lam$ be the boundary of $X_\tbar$. Let $X_{\sigma_i}\subset G/Q$ be the preimage of $X_{\sbar_i}$, then $$ \delp_\lam X_\tau^\kappa = \bigcup (X_\tau^\kappa\cap X_{\sigma_i}), $$ and this intersection holds scheme theoretically. In particular, the boundary is either empty or a pointed union of Richardson varieties. In view of Proposition \ref{emptylambdaboundary}, we shall henceforth consider only those $X_\tau^\kappa$ for which the $\lambda$-boundary is non-empty. \end{rem} It remains to define {\it standardness on $X_\tau^\kappa$\/} for the non-regular case: If $\pi=(\sigma_0,\ldots,\sigma_p,a_1,\ldots,a_p)$ is an L-S path of shape $\lam$, then $\pi$ is called {\it standard on\/ $X_\tau^\kappa$\/} if there exist {\it lifts} $\ts_0,\ldots,\ts_p\in W/W_Q$, i.e., $\ts_i\equiv \sigma_i\bmod W_\lam$ for all $i=0,\ldots,r$, such that $$ \tau\ge\ts_0\ge\ts_1\ge\ldots\ge\ts_p\ge\kappa. $$ Such a sequence $(\ts_0,\ldots,\ts_p)$ is called a {\it defining chain for $\pi$} (on $X_\tau^\kappa$). For the $\lambda$-boundary $\delp_\lam X_\tau^\kappa$ of the Richardson variety $X_\tau^\kappa$, let $\ci_{\delp_\lam X_\tau^\kappa}$ be the corresponding sheaf of ideals. \begin{thm}\label{idealfiltrationtwo} The twisted sheaf $\ci_{\delp_\lam X_\tau}(\lam)=\ci_{\delp_\lam X_\tau^\kappa}\otimes\cl_\lam$ admits a filtration, such that the associated graded is isomorphic (as $\co_{X_\kappa^\sigma}$--sheaf and as $T$-sheaf) to a direct sum of structure sheaves, twisted by a character: $$ \grad \ci_{\delp_\lam X_\tau^\kappa}(\lam) \simeq\bigoplus_{\pi\in S} \co_{X_{e(\pi)}^\kappa}\otimes\chi_{-\pi(1)}, $$ where $S$ is the set of all L--S paths $\pi$ of shape $\lam$, standard on $X_\tau^\kappa$ and such that $i(\pi)\equiv\tau\bmod W_\lam$. \end{thm} The rest of this section is devoted to the proof of Theorem \ref{idealfiltrationtwo}. For the rest of this section we fix a regular dominant weight $\rho\in\Lambda^{++}_Q$. Let $\pi\in B(\lam)$ be as above and let $\eta_1,\ldots,\eta_s,\eta_1',\ldots,\eta_q'\in B(\rho)$. \begin{defn}\rm A sequence $\ueta=(\eta_1,\ldots,\eta_s,\pi,\eta_1',\ldots,\eta_q')$ of shape $(s\rho,\lam,q\rho)$, is called {\it standard} if there exists a defining chain $(\ts_0,\ldots,\ts_p)$ for $\pi$ such that $$ e(\eta_1)\ge i(\eta_2)\ge\ldots\ge e(\eta_s)\ge \ts_0\ge\ldots\ge\ts_p\ge i(\eta'_{1})\ge e(\eta'_1) \ge \ldots\ge i(\eta'_q). $$ The sequence $\ueta$ is called {\it standard on $X_\tau^\kappa$} if in addition $\tau\ge i(\eta_1)$ (respectively $\tau\ge \ts_0$ if $s=0$) and $e(\eta'_q)\ge\kappa$ (respectively $\ts_p\ge \kappa$ if $q=0$). The sequence $(\ts_0,\ldots,\ts_p)$ is then called a {\it defining chain} for $\ueta$ on $X_\tau^\kappa$. \end{defn} The monomial $p_\ueta=p_{\eta_1}\cdots p_{\eta_s} p_\pi p_{\eta'_{1}}\cdots p_{\eta_q'}$ is called {\it standard} (on $X_\tau^\kappa$) if the sequence $\ueta$ is so. The monomial is called {\it standard on a union of Richardson varieties} if it is standard on at least one irreducible component. We have the following global basis given by standard monomials: \begin{thm}[\cite{Li$_2$}]\label{globalmonom} The standard monomials of shape $(s\rho,\lam,q\rho)$ form a basis for $H^0(G/Q,\cl_{\lam+(s+q)\rho})$ for all $s,q\ge 0$. \end{thm} The notion of the final and initial direction has to be adapted to standard sequences. Let $\ueta$ be a standard sequence on $X_\tau^\kappa$. The set of all defining chains for $\ueta$ on $X_\tau^\kappa$ is partially ordered: We say $(\ts_0,\ldots,\ts_p)\ge (\ts_0',\ldots,\ts_p')$ if and only if $\ts_0\ge\ts_0'$, $\ldots$, $\ts_p\ge \ts_p'$. One knows by Deodhar's Lemma that for this partial order there exists a unique minimal and a unique maximal defining chain (see for example \cite{$G/P$-IV}, \cite{$G/P$-V} or \cite{Li$_3$}) . \begin{defn}\label{nonregularlastchain}\rm If $s\ge 1$, then set $i(\ueta)=i(\eta_1)$ as before. If $s=0$, then set $i(\ueta)=\ts_0$ for the unique minimal defining chain $(\ts_0,\ldots,\ts_p)$. If $q\ge 1$, then set $e(\ueta)=e(\eta'_q)$. If $q=0$, then set $e(\ueta)=\ts_p$ for the unique maximal defining chain $(\ts_0,\ldots,\ts_p)$. \end{defn} \begin{rem}\label{definingandclassic}\rm If $q\ge 1$ or $s\ge 1$, then $i(\ueta)$ and $e(\ueta)$ are independent of the Richardson variety $X_\tau^\kappa$. If $q=s=0$, this is not the case: For $G=SL_3$, $Q=B$, $\lam=\omega_1$, $\eta=(id)$ and $X_\tau^\kappa=X_{id}^{s_1s_2}$, we have $i(\eta)=(id)$, but for $X_\tau^\kappa=X_{s_1}^{s_1s_2}$, we have $i(\eta)=(s_1)$. \end{rem} \begin{rem}\label{defininganddirect}\rm The path model theory provides for all $s,q\ge 0$, $q+s>0$, a natural weight preserving bijection between the set of all L--S paths $\pi$ of shape $\lam+(s+q)\rho$ such that $i(\pi)=\tau$ and $e(\pi)=\kappa$ and the set of standard sequences $\ueta$ of shape $(s\rho,\lam,q\rho)$ such that $i(\ueta)=\tau$ and $e(\ueta)=\kappa$, see \cite{Li$_3$}. \end{rem} \begin{prop}\label{smtonenonreg} The standard monomials of shape $(0,\lam,0)$ are linearly independent on a pointed union of Richardson varieties $Y$. Further, for all $s+q\ge 1$, the standard monomials on $Y$ of shape $(s\rho,\lam,q\rho)$ form a basis for $H^0(Y,\cl_{\lam+(s+q)\rho})$ and the corresponding higher cohomology groups vanish. \end{prop} {\it Proof.\/} Theorem~\ref{vanishandbasis} implies for $q+s\ge 1$ that the higher cohomology groups vanish, and, by Remark~\ref{defininganddirect}, the number of standard monomials, standard on $Y$, is equal to the dimension of $H^0(Y,\cl_{\lam+(s+q)\rho})$. So to prove the proposition in this case, it is sufficient to show that the standard monomials span the space of global sections. We consider only the case $q\ge1$, the case $s\ge 1$ can be proved similarly. The proof for a union of (opposite) Schubert varieties $X_\tau$ has been given in \cite{Li$_2$}. Suppose $Y=\bigcup_j X_{\tau_j}^\kappa$ is a pointed union of Richardson varieties. Denote by $Z$ the union of Schubert varieties $Z=\bigcup_j X_{\tau_j}$. The restriction map \begin{equation}\label{restrictionmapproposition} H^0(Z,\cl_{\lam+(q+s)\rho})\rightarrow H^0(Y,\cl_{\lam+(q+s)\rho}), \end{equation} is surjective by Theorem~\ref{vanishandbasis}. The standard monomials standard on $Y$ are also standard on $Z$. But a standard monomial $p_\ueta$, standard on $Z$, is not standard on $Y$ if and only if $e(\ueta)\not\ge\kappa$. But in this case the restriction of $p_\ueta\vert_Y$ vanishes ($q>0$!), so, by the surjectivity of $(\ref{restrictionmapproposition})$, the standard monomials, standard on $Y$, span the space of global sections. If $Y=\bigcup_j X_{\tau}^{\kappa_j}$, then let $Z$ be the Schubert variety $Z=X_{\tau}$. As above, the restriction map on the global sections is surjective, and the same arguments show that a standard monomial, standard on $Z$ but not on $Y$, vanishes identically on $Y$, which finishes the proof also in this case. It remains to consider the case $q=s=0$. Suppose first $Y=\bigcup_j X_{\tau_j}^\kappa$ is a pointed union of Richardson varieties and $p_{\pi_1},\ldots,p_{\pi_t}\in H^0(G/Q,\cl_\lam)$ are standard on $Y$, but $(\sum_{i=1}^t a_i p_{\pi_i})\vert_Y=0$. The sequence $(\kappa)$ is an L-S path of shape $\rho$, let $p_\kappa$ be the corresponding path vector. By the definition of standardness, the monomials $p_{\pi_i}p_\kappa$ of shape $(0,\lam,\rho)$ are standard on $Y$. It follows that the sum $\sum_{i=1}^t a_i p_{\pi_i}p_\kappa=0$ would be a linear dependence relation of standard monomials of shape $(0,\lam,\rho)$, in contradiction to what has been proved above. For $Y=\bigcup_j X_{\tau}^{\kappa_j}$ the arguments are similar, instead of $p_\kappa$ one takes the path vector $p_\tau\in H^0(G/Q,\cl_\rho)$ and deduces the linear independence of the $p_{\pi_1},\ldots,p_{\pi_t}$ by the linear independence of the $p_\tau p_{\pi_1},\ldots,p_\tau p_{\pi_t}$. \endpf We come now to the proof of Theorem~\ref{idealfiltrationtwo}. \par\noindent {\it Proof.\/} The exact sequence $0\rightarrow \ci_{\delp_\lam X_\tau^\kappa}\rightarrow \co_{X_\tau^\kappa}\rightarrow \co_{\delp_\lam X_\tau^\kappa}\rightarrow 0$ induces an exact sequence $$ 0\rightarrow \ci_{\delp_\lam X_\tau^\kappa}(\lam)\rightarrow \co_{X_\tau^\kappa}\otimes\cl_{\lam} \rightarrow \co_{\delp_\lam X_\tau^\kappa}\otimes\cl_{\lam}\rightarrow 0. $$ Since $\rho$ is regular, one gets for $m\gg 0$ an induced exact sequence \begin{equation}\label{shortexactnonregular} \begin{array}{rll} 0\rightarrow H^0(X_\tau^\kappa, \ci_{\delp_\lam X_\tau^\kappa}(\lam)\otimes \cl_{m\rho})\rightarrow &&\\ H^0(X_\tau^\kappa, \cl_{\lam+m\rho})&\rightarrow& H^0(\delp_\lam X_\tau^\kappa,\cl_{\lam+m\rho})\rightarrow 0. \end{array} \end{equation} Denote $S_\tbar^+$ {\it the set of L-S paths of shape $\lam$, standard on $X_\tau^\kappa$ and such that $\pi=(\tbar,\ldots)$}. This set is positive saturated, i.e., if $\pi,\pi'$ are elements of this set and $\pi'\ge\eta\ge\pi$, then $\eta$ is an element of this set too (see section~\ref{pathvector}). This set has a unique maximal element: the path $(\tbar)$. By deleting step by step one element, it is possible to get a sequence of subsets $$ S_0=\emptyset\subset S_1\subset\ldots \subset S_{N-1}\subset S_N= S_\tbar^+. $$ such that $(\tbar)\in S_j$ for all $j>0$ and the $S_j$ are positive saturated for all $j$. Fix $M\gg 0$ so that $(\ref{shortexactnonregular})$ holds for all $m\ge M$ and consider the $R$--module $$ I_\tbar^\kappa(\lam):=\bigoplus_{m\ge M} H^0(X_\tau^\kappa,\cl_{m\rho} \otimes \ci_{\delp_\lam X_\tau^\kappa}(\lam)). $$ The path vectors $p_\pi$, $\pi\in S^+_\tbar$, are sections in $H^0(X_\tau^\kappa,\cl_\lam)$ which vanish on $\delp_\lam X_\tau^\kappa$, this follows immediately from the description of the boundary in Remark~\ref{boundarydescription} and the fact that the $p_\pi$ vanishes on $\delp X_\tbar$. Set $$ (I_\tbar^\kappa(\lam))_j=\sum_{\pi\in S_j}\left(\sum_{m\ge M}H^0(X_\kappa^\tau,\cl_{m\rho}) p_{\pi}\right), $$ then $(I_\tbar^\kappa(\lam))_j$ is a $T$-stable $R$--submodule of $I_\tbar^\kappa(\lam)$ and hence this defines a filtration of $I_\tbar^\kappa(\lam)$ by $T$-stable $R$--submodules $$ 0\subset (I_\tbar^\kappa(\lam))_1\subset\ldots\subset (I_\tbar^\kappa(\lam))_N =I_\tbar^\kappa(\lam). $$ The middle term in the short exact sequence~{(\ref{shortexactnonregular})} has as basis the standard monomials of shape $(0,\lam,m\rho)$, standard on $X_\tau^\kappa$, and the last term has as basis the standard monomials of shape $(0,\lam,m\rho)$, standard on $\delp_\lam X_\tau^\kappa$ (see Proposition~\ref{smtonenonreg}). Let $I(\lam+m\rho)$ be the set of standard monomials of shape $(0,\lam,m\rho)$, standard on $X_\tau^\kappa$ but not standard on $\delp_\lam X_\tau^\kappa$. These are the standard monomials of the form $p_\ueta=p_\pi p_{\ueta'}$, $\pi\in S_\tbar^+$ and $p_{\ueta'}$ of shape $m\rho$. These monomials vanish on $\delp_\lam X_\tau^\kappa$. It follows that the $p_\ueta$, $\ueta\in I(\lam+m\rho)$, form a basis of the first term. The next step is to prove that the basis is compatible with the filtration. So suppose $p_\pi\in H^0(X_\tau^\kappa,\cl_\lam)$ is such that $\pi\in S_\tbar^+$ and let $p_\eta$ be a path vector in $H^0(X_\tau^\kappa,\cl_\rho)$ and standard on $X_\tau^\kappa$. Suppose the monomial $p_\pi p_\eta$ of shape $(0,\lam,\rho)$ is not standard on $X_\tau^\kappa$. Since $\rho\in\Lam^{++}$, this implies automatically that the product is not standard on the Schubert variety $X_\tau$. Now the quadratic relations in \cite{LiSe}, Theorem~3.13, show: $$ p_\pi p_\eta = \sum_{\pi',\eta'} b_{\pi',\eta'} p_{\pi'}p_{\eta'} $$ gives a presentation of the product as linear combination of standard monomials, standard on $X_\tau$, where $(\pi',\eta')\ge(\pi,\eta)\ge^r (\pi',\eta')$. Such a standard monomial $p_\pi p_\eta$ is then not standard on $X_\tau^\kappa$ if and only if $e(\eta')\not\ge\kappa$, in which case the monomial vanishes on $X_\tau^\kappa$. But this implies that we have the same type of relation for non-standard products also on $X_\tau^\kappa$ and hence: \begin{equation} p_\pi p_\eta = \sum_{(\pi',\eta')\ge(\pi,\eta)\ge^r (\pi',\eta')} b_{\pi',\eta} p_{\pi'}p_{\eta'}, \hbox{\rm\ all\ $p_{\pi'}p_{\eta'}$\ are standard on\ }X_\tau^\kappa \end{equation} The sets $S_j$ are positively saturated, and hence (as in the regular case) the subspace spanned by elements of degree $m$ in the $j$--th filtration part: $$ (I_{\tbar,m}^\kappa(\lam))_j=H^0(X_{\tau}^\kappa, \ci_{\delp_\lam X_\tau^\kappa}(\lam)\otimes \cl_{m\rho})_j $$ has a basis given by the standard monomials, standard on $X_\tau^\kappa$, starting with a $p_\pi$, $\pi\in S_j$. Let $\{\pi\}= S_j-S_{j-1}$. As in the regular case, the map $$ (I_{\tbar,m}^\kappa(\lam))_j\rightarrow H^0(X_{e(\pi)}^\kappa,\cl_{m\rho}) \qquad \left(p_\pi f_\pi+\sum_{\eta\in S_{j-1}} p_{\eta} f_\eta\right)\mapsto f_\pi\vert_{X_{e(\pi)}^\kappa}. $$ induces an isomorphism $(I_{\tbar,m}^\kappa(\lam))_j/(I_{\tbar,m}^\kappa(\lam))_{j-1}\longrightarrow H^0(X_{e(\pi)}^\tau,\cl_{m\rho})$. Since this holds for all $m\ge M$, this induces a filtration of the sheaf $$ 0\subset (\ci_{\delp_\lam X_\tau^\kappa}(\lam))_1\subset (\ci_{\delp_\lam X_\tau^\kappa}(\lam))_2\subset \ldots\subset (\ci_{\delp_\lam X_\tau^\kappa}(\lam))_N=\ci_{\delp_\lam X_\tau^\kappa}(\lam) $$ such that the associated graded is isomorphic to the direct sum $$ \grad \ci_{\delp_\lam X_\tau^\kappa}(\lam)=\bigoplus_{\pi\in S} \co_{X_{e(\pi)}^\kappa}\otimes\chi_{-\pi(1)}. $$ \endpf \section{SMT in the non-regular case}\label{SMTnonreg} We have the following generalization of Theorem~\ref{vanishandbasis} to the non-regular case: \begin{thm}\label{vanishandbasisnonreg} Suppose $\lam\in\Lambda^+_Q$ is a dominant weight and let $Y$ be a pointed union of Richardson varieties. \begin{joliste} \item[{\rm (i)}] $H^i(Y,\cl_{m\lam})=0$ for all $i\ge 1$ and all $m\ge 1$. \item[{\rm (ii)}] The standard monomials, standard on $Y$ of degree $m$, form a basis for $H^0(Y,\cl_{m\lam})$ for all $m\ge 1$. \end{joliste} \end{thm} {\it Proof.} We give only a sketch of the proof for $m=1$ (respectively $m=0$ in {\it i)}, the case $m\ge 2$ can be proved in the same way as in the regular case. The proof of the appropriate version of Proposition~\ref{smtonenonreg} for monomials of type $(s\rho,m\lam,q\rho)$ is the same. The proof is (as in the regular case) by induction on the number of components of maximal dimension and on the dimension. The theorem holds obviously in case $Y$ is a point. We assume now that the theorem holds for all pointed unions of Richardson varieties of dimension smaller than $n$. Suppose $X_\kappa^\sigma$ is of dimension $n$ and consider the projection $\phi: X_\kappa^\sigma \longrightarrow G/P_\lam$. If $\delp_\lam X_\kappa^\sigma=\emptyset$, then $X_\kappa^\sigma\subset p^{-1}(e_{\kbar})$ by Proposition~\ref{emptylambdaboundary}, and the restriction of the line bundle to $X_\kappa^\sigma$ is trivial. The theorem holds obviously in this case. If $\delp_\lam X_\kappa^\sigma\not=\emptyset$, then the inclusion $\ci_{\delp_\lam X_\kappa^\sigma} \hookrightarrow\co_{X_\kappa^\sigma}$, tensored by $\cl_\lam$, induces an exact sequence $$ 0\rightarrow \ci_{\delp_\lam X_\kappa^\sigma}(\lam) \rightarrow\co_{X_\kappa^\sigma}\otimes\cl_\lam \rightarrow \co_{\delp_\lam X_\kappa^\sigma}\otimes\cl_\lam\rightarrow 0. $$ By Theorem~\ref{idealfiltrationtwo}, the sheaf $\ci_{\delp_\lam X_\kappa^\sigma}(\lam)$ admits a filtration such that the associated graded is isomorphic to $\bigoplus \co_{X_{e(\pi)}^\sigma} \otimes \chi_{-\pi(1)}$. Since $\delp_\lam X_\kappa^\sigma$ is a pointed union of Richardson varieties of smaller dimension, the vanishing theorem for higher cohomology groups holds. The filtration in Theorem~\ref{idealfiltrationtwo} and the vanishing in Theorem~\ref{vanishandbasis}(i) shows that $\ci_{\delp_\lam X_\kappa^\sigma}(\lam)$ has vanishing higher cohomology groups, so the long exact sequence shows that $H^i(X_\kappa^\sigma,\cl_\lam)=0$ for all $i\ge 1$. Further, the short exact sequence $$ 0\rightarrow H^0(X_\kappa^\sigma,\ci_{\delp_\lam X_\kappa^\sigma}(\lam)) \rightarrow H^0(X_\kappa^\sigma,\cl_\lam) \rightarrow H^0(\delp_\lam X_\kappa^\sigma,\cl_\lam)\rightarrow 0 $$ shows $\dim H^0(X_\kappa^\sigma,\cl_\lam)=\dim H^0(\delp_\lam X_\kappa^\sigma,\cl_\lam)+ \dim H^0(X_\kappa^\sigma,\ci_{\delp_\lam X_\kappa^\sigma}(\lam))$. Consider the right hand side of the equation. By induction, the first term is equal to the number of L-S paths of shape $\lam$ standard on $\delp_\lam X_\kappa^\sigma$, and the second is by the filtration equal to the number of L-S paths of shape $\lam$ standard on $X_\kappa^\sigma$ and such that $i(\pi)=\kappa$. This together is the number of L-S paths of shape $\lam$ standard on $X_\kappa^\sigma$. Since we have already proved that the restrictions of the corresponding path vectors remain linearly independent (Proposition~\ref{smtonenonreg}), this proves that the path vectors form a basis of $H^0(X_\kappa^\sigma,\cl_\lam)$. Suppose now $Y$ is a pointed union of Richardson varieties. Let $X_\kappa^\sigma$ be an irreducible component of maximal dimension and denote $Y$ the union of the remaining irreducible components. We have an exact sequence: $$ 0\rightarrow \co_{Y}\rightarrow \co_{X_\kappa^\sigma}\oplus \co_{Y'}\rightarrow \co_{Y\cap X_\kappa^\sigma}\rightarrow 0. $$ If we tensor the sequence with $\cl_{\lam}$, then we know by induction on the dimension, respectively the number of irreducible components of maximal dimension that the higher cohomology groups of the second and the last term vanish, and further, for the global sections the last map is surjective. It follows that the higher cohomology groups $H^i(Y,\cl_{\lam})$ vanish. It remains to count the dimensions of the spaces of global sections. Since one has standard monomial theory for the last two terms by induction on the dimension, respectively the number of irreducible components of maximal dimension, one checks easily that in fact $\dim H^0(Y,\cl_{\lam})$ is equal to the number of standard monomials on $Y$. The linear independence (Proposition~\ref{smtonenonreg}) then implies that the standard monomials form a basis of $\dim H^0(Y,\cl_{\lam})$. \endpf \begin{rem}\rm The result (i) in Theorem \ref{vanishandbasisnonreg} is also proved in \cite{brionlak} (cf. \cite{brionlak}, Proposition 1) \end{rem} \section{Standard monomials and equivariant $K$-theory}\label{SMTequivariantKtheory} In this section we will show that the notion of a ``standard monomial'' is directly related to geometry of Richardson varieties, and this can be best expressed in terms of equivariant $K$-theory. The Chow ring Chow$(G/B)$ has a $\bz$-basis consisting of $[X_\tau]$, the class in Chow$ (G/B)$ defined by the cycle represented by the Schubert variety $X_\tau$, $\tau \in W$. Let $\lambda \in \Lambda$ (=the weight lattice), $\cl_\lambda$ the associated line bundle. The class $[\cl_{\lambda}]$ is defined by a codimension one cycle. We have, the following expression for the product $[X_\tau] \cdot [\cl_\lambda]$ in the Chow ring: $$ [X_\tau] \cdot [\cl_\lambda]= \sum a_i [X_{\tau_i}]\leqno(1) $$ where the summation runs over all Schubert divisors $X_{\tau_i}$ in $X_\tau$; further, one has an explicit expression for $a_i$ due to Chevalley (cf. \cite{che}). When $G = SL(n)$, (1) is just the classical formula of Pieri. Denote by $K(G/B)$ the Grothendieck ring of (isomorphism classes of) coherent sheaves on $G/B$. Then one knows that the classes $[{\mathcal{O}}_{X_\tau}]$ in $K(G/B)$ of the structure sheaves ${\mathcal{O}}_{X_\tau}$ of the Schubert varieties $X_\tau$ form a $\bz$-basis of $K(G/B)$. Let $$ [{\mathcal{O}}_{X_\tau}]\cdot [\cl_{\lambda}] = \sum_{\kappa \in W} a_{\tau, \kappa}^{\lambda} [{\mathcal{O}}_{X_\kappa}]\leqno(2) $$ More generally, let $K_T (G/B)$ be the Grothendieck ring of $T$-equivariant sheaves on $G/B$. By \cite{KK} one knows again that the classes $[{\mathcal{O}}_{X_\tau}]_T$ of the structure sheaves of the Schubert varieties form a $\bz [\Lambda ]$-basis, where ${\bz} [\Lambda ]$ is the group algebra of the weight lattice $\Lambda$. Let $$ [{\mathcal{O}}_{X_\tau}]_T\cdot [\cl_{\lambda}]_T = \sum_{\kappa \in W} C_{\tau,\kappa}^{\lambda} [{\mathcal{O}}_{X_\kappa}]_T\leqno(3) $$ where $C_{\tau,\kappa}^{\lambda}$'s are formal sums of characters of $T$ with integral coefficients. Assume now that $\lambda \in \Lambda^+$. Then one knows that in (1) all $a_i \geq 0$ (cf.\cite{che}). The integers $a_{\tau, \kappa}^\lambda$ in (2) were determined by Fulton and Lascoux (cf. \cite{FL}) for the case $G = SL(n)$ ; they provide a formula for $a_{\tau, \kappa}^{\lambda}$ (using the combinatorics of Grothendieck polynomials). The general case was treated by Mathieu using representation theory (cf. \cite{M}), who shows that $C_{\tau,\kappa}^{\lambda}$ are effective i.e., formal sums of characters with positive integral coefficients. For a L-S path $\pi=(\tau_1> \cdots>\tau_r;a_1<\cdots \sigma_i$ implies $j>i$. Set $Y^j=\bigcup_{1\le i\le j} X^\kappa_{\sigma_i}\times X^{\sigma_i}_\tau$, so we get a filtration $$ Y^1=X^\kappa_{\kappa}\times X^{\kappa}_\tau\subset Y^2\subset\ldots\subset Y^q=Y. $$ Set $\cR_Y(j)=\bigoplus_{n\ge 0} H^0(Y^j,\cl_{n\lam}\otimes \cl_{n\lam})$, this ring again has a basis consisting of standard monomials. Set $K^{1}=\cR_Y$, and, for $j=1,\ldots,q$, let $K^{j+1}\subset \cR_Y$ be the kernel of the restriction map $\cR_Y\rightarrow \cR_Y(j)$. We get a filtration $$ K^{q+1}=0\subset K^{q}\subset \ldots\subset K^2\subset K^{1}=\cR_Y. $$ The kernels $K^j$ have again a basis by standard monomials. In fact, recall the basis elements $p_{\upi_1}\otimes p_{\upi_2}$ in $SM_\tau^\kappa(\lam)$, then $K^j$ is spanned by those such that $i(\upi_1)=\sigma_i$ for some $i\ge j$. Consider the map $$ K^j\rightarrow H^0(X^\kappa_{\sigma_j}\times X^{\sigma_j}_\tau,\cl_{n\lam}\otimes \cl_{n\lam}), \quad s\mapsto s\vert_{X^\kappa_{\sigma_j}\times X^{\sigma_j}_\tau}. $$ It follows immediately that $K^{j+1}$ is the kernel of this map, and, by the proof of Theorem~\ref{k-smt}, we get an isomorphism of $\cR$--modules: $$ K^j/K^{j+1}\simeq \bigoplus_{n\ge 0} H^0(X^\kappa_{\sigma_j},\ci_{\delp X^\kappa_{\sigma_j}}(n\lam)) \otimes H^0(X^{\sigma_j}_\tau,\cl_{n\lam}) $$ Since all these maps are $T$-equivariant, translated into the language of sheaves of $\co_Z$--modules, this means that the filtration of $\cR_Y$ induces a $T$--stable filtration of $\co_Y$ as $\co_Z$--sheaf such that the associated graded is a direct sum of the $T$--equivariant $\co_Z$--sheaves $\ci_{\delp X^\kappa_{\sigma_j}}\otimes \co_{X^{\sigma_j}_\tau}$. The proof for the second filtration is similar and is left to the reader. To describe the flat family note that the kernel of the product map $$ \cR_{\tau}^\kappa\rightarrow \cR_{\mbox{$\scriptstyle \Delta X_\tau^\kappa$}} =\bigoplus_{n\ge 0} H^0(\Delta X_\tau^\kappa,\cl_{n\lam}\otimes\cl_{n\lam}) $$ is the ideal generated by the commutation relations $p_{\pi_1}\otimes p_{\pi_2}-p_{\pi_2}\otimes p_{\pi_1}$ for all $\pi_1,\pi_2\in B_\tau^\kappa(\lam)$, and if neither $p_{\pi_1}p_{\pi_2}$ nor $p_{\pi_2}p_{\pi_1}$ is standard in $H^0(X_\tau^\kappa,\cl_{2\lam})$, then we have the additional relations $$ p_{\pi_1}\otimes p_{\pi_2} -\sum a_{\eta_1,\eta_2} p_{\eta_1}\otimes p_{\eta_2} $$ where the $p_{\eta_1}p_{\eta_2}$ are standard and the coefficients $a_{\eta_1,\eta_2}\not=0$ only if $(\eta_1,\eta_2)\succeq \pi_1\wedge\pi_2\succeq^r (\eta_1,\eta_2)$. This follows easily from Theorem~\ref{quadraticrelation}, using a monomial order as in Proposition~\ref{homcoordinate}. Further, these elements form a reduced Gr\"obner basis for the ideal. The existence of the flat deformation follows now from standard Gr\"obner basis arguments. \begin{rem}\rm The existence of a filtration and a flat family (as in Theorem \ref{k-smt1}) is also proved in \cite{brionlak}. The existence of a filtration is proved in \cite{brion} for any Cohen-Macaulay subvariety $X$ of a flag variety, in general position with respect to opposite Schubert varieties (note that the assumptions on $X$ hold for Schubert varieties) . 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