%% if you are submitting an initial manuscript then you should have submission as an option here %% if you are submitting a revised manuscript then you should have revision as an option here %% otherwise options taken by the article class will be accepted \documentclass[finalversion]{FPSAC2024} \articlenumber{37} %% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc %% note that the class file already loads {amsmath, amsthm, amssymb} \usepackage{comment} \usepackage{enumitem} \usepackage{amsfonts} \usepackage{mathtools} \usepackage{hyperref} \usepackage{dynkin-diagrams} \usepackage[utf8]{inputenc} \usepackage{tikz,tikz-cd} \newcommand{\Red}{\mathrm{Red}} \newcommand{\C}{\mathcal{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\bigdot}{\boldsymbol{\cdot}} \newcommand\precdot{\mathrel{\ooalign{$\prec$\cr \hidewidth\raise0.001ex\hbox{$\cdot\mkern0.6mu$}\cr}}} \newtheorem{theorem}{Theorem}[section] \newtheorem{def-prop}[theorem]{Definition-Proposition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{claim}[theorem]{Claim} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{problem}[theorem]{Problem} \theoremstyle{definition} \newtheorem{ex}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem*{remark}{Remark} %% define your title in the usual way \title[Levi-spherical varieties and Demazure characters]{Levi-spherical varieties and Demazure characters} %% define your authors in the usual way %% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details \author{Yibo Gao\thanks{\href{mailto:gaoyibo@bicmr.pku.edu.cn}{gaoyibo@bicmr.pku.edu.cn}}\addressmark{1}, Reuven Hodges\thanks{\href{mailto:rmhodges@ku.edu}{rmhodges@ku.edu}. RH was partially supported by an AMS-Simons Travel Grant.}\addressmark{2}, \and Alexander Yong\thanks{\href{mailto:ayong@illinois.edu}{ayong@illinois.edu}. AY was partially supported a Simons Collaboration Grant and an NSF RTG.}\addressmark{3}} %% then use \addressmark to match authors to institutions here \address{\addressmark{1}Beijing International Center for Mathematical Research, Peking University, Beijing, China\\ \addressmark{2}Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA \\ \addressmark{3}Department of Mathematics, U.~Illinois at Urbana-Champaign, Urbana, IL 61801, USA} %% put the date of submission here \received{11/14/2023} %% leave this blank until submitting a revised version \revised{3/27/2024} %% put your English abstract here, or comment this out if you don't have one yet %% please don't use custom commands in your abstract / resume, as these will be displayed online %% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year) \abstract{We prove a short, root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety $G/B$ of finite Lie type. We apply this to the study of multiplicity-free decompositions of Demazure modules and their characters.} %% put your French abstract here, or comment this out if you don't have one %\resume{\lipsum[2]} %% put your keywords here, or comment this out if you don't have them yet \keywords{Demazure characters, multiplicity-free, Schubert varieties, Levi subgroups, spherical varieties, toric varieties} %% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier %% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text %% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex %% but you can use bibtex if you want \usepackage[backend=bibtex]{biblatex} \addbibresource{GHY.FPSAC2024.bib} %% note the \printbibliography command at the end of the file which goes with these biblatex commands \begin{document} \maketitle %% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above \section{Introduction} \subsection{History and motivation} In his essay \cite{Howe} on representation theory and invariant theory, R. Howe discusses the significance of multiplicity-free actions as an organizing principle for the subject. Classical invariant theory focuses on actions of a reductive group $G$ on symmetric algebras, which is to say, coordinate rings of vector spaces. Now one also considers $G$-actions on varieties $X$ and their coordinate rings ${\mathbb C}[X]$. Such an action is multiplicity-free if ${\mathbb C}[X]$ decomposes, as a $G$-module, into irreducible $G$-modules each with multiplicity one. An important example is when $X$ is the \emph{base affine space} of a complex, semisimple algebraic group $G$ \cite{BGG75}; in this case the coordinate ring is a multiplicity-free direct sum of the irreducible representations of $G$. Lustzig's theory of dual canonical bases \cite{L90} provides a basis for it. In the early 2000s, understanding this basis was a motivation for S.~Fomin and A.~Zelevinsky's theory of Cluster algebras \cite{FZ02}. The notion of multiplicity-free actions is closely connected to that of \emph{spherical varieties}. Let $G$ be a connected, complex, reductive algebraic group; we say that a variety $X$ is a $G$-variety if $X$ is equipped with an action of $G$ that is a morphism of varieties. A spherical variety is a normal $G$-variety where a Borel subgroup of $G$ has an open, and therefore dense, orbit. A normal, affine $G$-variety $X$ is spherical if and only if ${\mathbb C}[X]$ decomposes into irreducible $G$-modules each with multiplicity one \cite{VK78}. If $X$ is instead a normal, projective $G$-variety then one can still recover one direction of this implication. That is, if the induced $G$-action on the homogeneous coordinate ring of $X$ is multiplicity-free, then $X$ is $G$-spherical \cite[Proposition 4.0.1]{Hodges.Lakshmibai}. Spherical varieties possess numerous nice properties. For instance, projective spherical varieties are Mori Dream Spaces. Moreover, Luna-Vust theory describes all the birational models of a spherical variety in terms of colored fans; these fans generalize the notion of fans used to classify toric varieties (which are themselves spherical varieties). N.~Perrin's excellent survey covers additional background on spherical varieties \cite{Perrin}. It is an open problem to classify all spherical actions on products of flag varieties. This is solved in the case of Levi subgroups; we point to the work of P.~Littelmann \cite{Littel}, P.~Magyar--J.~Weyman--A.~Zelevinsky \cite{MWZ, MWZ2}, J.~Stembridge \cite{STEM,Stem:Weyl}, R.~Avdeev--A.~Petukhov \cite{AP14, AP20}. Connecting back to the representation-theoretic perspective of \cite{Howe}, in \cite{STEM,Stem:Weyl}, J.~Stembridge relates this classification problem to the multiplicity-freeness of restrictions of \emph{Weyl modules} \cite[Lecture~6]{Fulton.Harris}. Indeed, the homogeneous coordinate ring of a single flag variety is a multiplicity-free sum of spaces of global sections on the variety with respect to line bundles associated to each dominant integral weight. By the Borel-Weil-Bott theorem, these spaces are isomorphic to the irreducible representations of $G$. This is the central object of interest in \emph{Standard Monomial Theory} \cite{LR08} and is closely related to the coordinate ring of base affine space mentioned above. As remarked above a product of flag varieties is $G$-spherical if its homogeneous coordinate ring is multiplicity-free as an $G$-module. This paper solves a related problem. We classify all \emph{Levi-spherical} Schubert varieties in a single flag variety; that is, Schubert varieties that are spherical for the action of a Levi subgroup. Here, the relevant ring is the homogeneous coordinate ring of a Schubert variety and the attendant representation theory is that of \emph{Demazure modules} \cite{Demazure}, which are Borel subgroup representations. Critically for this paper, they are also Levi subgroup representations. Multiplicity-freeness in this setting refers to the decomposition of these modules into irreducible Levi subgroup representations. This study was initiated in \cite{Hodges.Yong1} and the authors solved the problem for the $GL_n$ case in \cite{GHY}. In \cite{GHY2} we conjectured an answer for all finite rank Lie types; this paper proves that conjecture.\footnote{During the completion of this article, we learned that M.~Can-P.~Saha \cite{CanSaha} independently proved the conjecture.} \subsection{Background} Throughout, let $G$ be a complex, connected, reductive algebraic group and let $B\leq G$ be a choice of Borel subgroup along with a maximal torus $T$ contained in $B$. The \emph{Weyl group} is $W:= N_G(T)/T$, where $N_G(T)$ is the normalizer of $T$ in $G$. The orbits of the homogeneous space $G/B$ under the action of $B$ by left translations are the \emph{Schubert cells} $X_w^{\circ}, w\in W$. Their Zariski closures \[X_w:=\overline{X_w^{\circ}}\] are the \emph{Schubert varieties}. It is relevant below that these varieties are normal \cite{DeCon-Lak, RR}. A \emph{parabolic subgroup} of $G$ is a closed subgroup $B\subset P\subsetneq G$ such that $G/P$ is a projective variety. Each such $P$ admits a \emph{Levi decomposition} \[P=L\ltimes R_u(P)\] where $L$ is a reductive subgroup called a \emph{Levi subgroup} of $P$ and $R_u(P)$ is the unipotent radical. One parabolic subgroup is $P_w:={\rm stab}_G(X_w)$. Any of the parabolic subgroups $B\subseteq Q\subseteq P_w$ act on $X_w$. Let $L_Q$ be a Levi subgroup of $Q$. A variety $X$ is \emph{$H$-spherical} for the action of a complex reductive algebraic group $H$ if it is normal and contains an open, and therefore dense, orbit of a Borel subgroup of~$H$. Our reference for spherical varieties is \cite{Perrin}; toric varieties are examples of spherical varieties. \begin{definition}[{\cite[Definition~1.8]{Hodges.Yong1}}] Let $B\subseteq Q\subseteq P_w$ be a parabolic subgroup of $G$. $X_w\subseteq G/B$ is \emph{$L_Q$-spherical} if $X_w$ has a dense, open orbit of a Borel subgroup of $L_Q$ under left-translations. \end{definition} \subsection{The main result} We give a root-system uniform combinatorial criterion to decide if $X_w$ is $L_Q$-spherical. Let $\Phi:=\Phi({\mathfrak g},T)$ be the root system of weights for the adjoint action of $T$ on the Lie algebra ${\mathfrak g}$ of $G$. It has a decomposition $\Phi=\Phi^+\cup \Phi^-$ into positive and negative roots. Let $\Delta\subset \Phi^+$ be the base of simple roots. The parabolic subgroups $Q=P_I\supset B$ are in bijection with subsets $I$ of $\Delta$; let $L_I:=L_Q$. The set of \emph{left descents} of $w$ is \[{\mathcal D}_L(w)=\{\beta\in \Delta: \ell(s_{\beta}w)<\ell(w)\},\] where $\ell(w)=\dim X_w$ is the \emph{Coxeter length} of $w$. Under the bijection, $P_w=P_{{\mathcal D}_L(w)}$, and $B\subset Q\subseteq P_w=P_{{\mathcal D}_L(w)}$ satisfy $Q=P_I$ for some $I\subseteq {\mathcal D}_L(w)$. For $I\subset \Delta$, a \emph{parabolic subgroup} $W_I\subseteq W$ is the subgroup generated by $S_I:=\{s_{\beta}:\beta \in I\}$. A \emph{standard Coxeter element} $c\in W_I$ is any product of the elements of $S_I$ listed in some order. Let $w_0(I)$ be the longest element of $W_I$. The following definition was given in type $A$ in \cite[Definition~1.1]{GHY} and in general type in \cite[Section~4]{GHY2}: %[{\cite[Definition~1.1]{GHY}, \cite[Section~4]{GHY2}}] \begin{definition} Let $w\in W$ and $I\subseteq {\mathcal D}_L(w)$ be fixed. Then $w$ is $I$-spherical if $w_0(I)w$ is a standard Coxeter element for $W_{J}$ where $J\subseteq \Delta$. \end{definition} We first note that if $I\subseteq {\mathcal D}_L(w)$, then the left inversion set ${\mathcal I}(w)$, defined in Section~\ref{sec:2}, contains all the positive roots in the root subsystem generated by $I$, and thus $w=w_0(I)d$ is a length-additive expression for some $d\in W$, by Proposition 3.1.3 in \cite{Bjorner.Brenti}. \begin{theorem} \label{conj:main} Fix $w\in W$ and $I\subseteq {\mathcal D}_L(w)$. $X_w$ is $L_I$-spherical if and only if $w$ is $I$-spherical. \end{theorem} Theorem~\ref{conj:main} resolves the main conjecture of the authors' earlier work \cite[Conjecture~4.1]{GHY2} and completes the main goal set forth in \cite{Hodges.Yong1}. In \cite{GHY}, Theorem~\ref{conj:main} was established in the case $G=GL_n$ using essentially algebraic combinatorial methods concerning \emph{Demazure characters} (or in their type $A$ embodiment, the \emph{key polynomials}). In contrast, the geometric arguments of this paper are quite different, significantly shorter, but require more background of the reader in algebraic groups. Theorem~\ref{conj:main} is a generalization of work of P.~Karuppuchamy \cite{Karrup} that characterizes Schubert varieties that are toric, which in our setup means spherical for the action of $L_{\emptyset}=T$. Using work of R.~S.~Avdeev--A. V. Petukhov \cite{AP14}, Theorem~\ref{conj:main} may also be seen as a generalization of some results of P.~Magyar--J.~Weyman--A.~Zelevinsky \cite{MWZ} and J.~Stembridge~\cite{STEM, Stem:Weyl} on spherical actions on $G/B$; see \cite[Theorem~2.4]{Hodges.Yong1}. Previously, there was not even a finite algorithm to decide $L_I$-sphericality of $X_w$ in general. \subsection{Organization} Examples of the main result are given in Section~\ref{sec:5}. Sections~\ref{sec:2} and~\ref{sec:3} prove Theorem~\ref{conj:main}. Section~\ref{sec:4} applies our main result to the study of Demazure modules \cite{Demazure}. \section{Examples of Theorem~\ref{conj:main}}\label{sec:5} We begin with a few examples illustrating Theorem~\ref{conj:main}. \begin{ex}[$E_8$ {cf.~\cite[Example~1.3]{Hodges.Yong1}}] The $E_8$ Dynkin diagram is \dynkin[label]E8. One associates the simple roots $\beta_i$ ($1\leq i\leq 8$) with this labeling and writes $s_i:=s_{\beta_i}$. Suppose \[w=s_2 s_3 s_4 s_2 s_3 s_4 s_5 s_4 s_2 s_3 s_1 s_4 s_5 s_6 s_7 s_6 s_8 s_7 s_6 \in W.\] Then ${\mathcal D}_L(w)=\{\beta_2,\beta_3,\beta_4,\beta_5,\beta_7,\beta_8\}$. Let $I={\mathcal D}_L(w)$. Here \[w_0(I)=s_3 s_2 s_4 s_3 s_2 s_4 s_5 s_4 s_3 s_2 s_4 s_5 \cdot s_7 s_8 s_7 \text{\ \ and \ \ } w_0(I)w=s_1 s_6 s_7 s_8.\] Since $w=w_0(I)c$ where $c=s_1 s_6 s_7 s_8$ is a standard Coxeter element, Theorem~\ref{conj:main} asserts that $X_w$ is $L_I$-spherical in the complete flag variety for $E_8$. \end{ex} \begin{ex}[$F_4$ {cf.~\cite[Example~1.5]{Hodges.Yong1}}] The $F_4$ diagram is \dynkin[label, edge length=0.5cm]F4. First suppose \[w=s_4 s_3 s_4 s_2 s_3 s_4 s_2 s_3 s_2 s_1 s_2 s_3 s_4 \text{\ ($I={\mathcal D}_L(w)=\{\beta_2,\beta_3,\beta_4\}$)}.\] Then $w_0(I)= s_2 s_3 s_2 s_3 s_4 s_3 s_2 s_3 s_4$ and $w_0(I)w=s_1 s_2 s_3 s_4$ is standard Coxeter. Hence $X_w$ is $L_I$-spherical. On the other hand if \[w'=s_2 s_1 s_4 s_3 s_2 s_1 s_3 s_2 s_4 s_3 s_2 s_1 \text{\ ($I={\mathcal D}_L(w')=\{\beta_2,\beta_4\}$)},\] then $w_0(I)=s_2 s_4$ and $w_0(I)w=s_1 s_3 s_2 s_1 s_3 s_2 s_4 s_3 s_2 s_1$ is not standard Coxeter and $X_w$ is not $L_{I}$-spherical. \end{ex} \begin{ex}[$D_4$] The $D_4$ diagram is \dynkin[label, edge length=0.5cm]D4. Let \[w= s_3 s_2 s_3 s_4 s_2 s_1 s_2 \text{\ ($I={\mathcal D}_L(w)=\{\beta_2,\beta_3\}$).}\] Thus $w_0(I)=s_2 s_3 s_2$ and $w_0(I)w=s_4 s_2 s_1 s_2$ is not standard Coxeter. Hence $X_w$ is not $L_I$-spherical. The interested reader can check $w$ is $I$-spherical in the (different) sense of \cite[Definition~1.2]{Hodges.Yong1}. Therefore, this $w$ provides a counterexample to \cite[Conjecture~1.9]{Hodges.Yong1} in type $D_4$. This counterexample was also (implicitly) verified in \cite{GHY2} using a different method, namely Demazure character computations, the topic of Section~\ref{sec:4}. \end{ex} \section{An equivariant isomorphism}\label{sec:2} The primary goal of this section is to construct a torus equivariant isomorphism from a specified affine subspace of the open cell of a Schubert variety to the open cell of a distinguished Schubert subvariety. In what follows, we assume standard facts from the theory of algebraic groups. References we draw upon are \cite{Humphreys, B91, LR08}. Let $w \in W$. Let $n_w$ be a coset representative of $w$ in $N_G(T)$. By definition of $N_G(T)$ being the normalizer of $T$ in $G$, $t \mapsto n_w t n_w^{-1}$ defines an automorphism $\gamma_w:T\to T$. \begin{lemma} \label{lemma:wellDefinedt_w} The automorphism $\gamma_w$ does not depend on our choice of coset representative $n_w$. \end{lemma} % \begin{proof} % Suppose that $m_w$ is another coset representative of $w$. Then $m_w = n_w s$ for some $s \in T$. Hence $m_w t m_w^{-1} = n_w s t s^{-1} n_w^{-1} = n_w t s s^{-1}n_w^{-1}=n_w t n_w^{-1}$. % \end{proof} In light of Lemma~\ref{lemma:wellDefinedt_w}, henceforth for $w \in W$ we will also let $w$ denote a coset representative of $w$ in $N_G(T)$. Let $X$ be a $T$-variety with action denoted by $\bigdot$. For each $w \in W$ we define an action $\bigdot_w$ on $X$ by $t \bigdot_w x = \gamma_w(t) \bigdot x$ for all $x \in X$ and $t \in T$. \begin{lemma} \label{lemma:denseByAutomorphism} For all $w \in W$, the $T$-variety $X$ has an open, dense $T$-orbit for the action $\bigdot$ if and only if it has an open, dense $T$-orbit for the action $\bigdot_w$. Indeed, the set of $T$-orbits in $X$ for these two actions is identical. \end{lemma} % \begin{proof} % Let $\mathcal{O}$ be a $T$-orbit in $X$ for the $\bigdot$ action. Let $x,y \in \mathcal{O}$ and $t \in T$ be such that $t \bigdot x = y$. As $\gamma_w$ is an automorphism, there exists a $t^{\prime} \in T$ such that $\gamma_w(t^{\prime}) = t$. Then % \[t^{\prime} \bigdot_w x = \gamma_w (t^{\prime}) \bigdot x = t \bigdot x = y.\] % Thus $\mathcal{O}$ is contained in the $T$-orbit $\mathcal{O}'$ of $x$ for the action $\bigdot_w$. % The reverse containment is true by definition of $\bigdot_w$. % The lemma % follows. % \end{proof} For the remainder, we fix $\bigdot$ to be the restriction to $T$ of the action of $G$ on $G/B$ by left multiplication. The \emph{left inversion set} of $w \in W$ is \[{\mathcal I}(w):=\Phi^+ \cap w(\Phi^-) = \{ \alpha \in \Phi^+ | w^{-1}(\alpha) \in \Phi^- \}.\] Recall two standard facts regarding left inversion sets \cite[Chapter 1]{Humphreys:grey}. For $w \in W$, \begin{equation} \label{eqn:leftinvIdent1} |{\mathcal I}(w)| = \ell(w) = \dim_{\mathbb C} X_w, \end{equation} and \begin{equation} \label{eqn:leftinvIdent2} {\mathcal I}(w_0(I)) = \Phi^+(I), \end{equation} where $\Phi(I)=\Phi({\mathfrak l}_I,T)$ is the root system for the adjoint action of $T$ on ${\mathfrak l}_I={\rm Lie}(L_I)$. We say that an algebraic group $H$ is \emph{directly spanned} by its closed subgroups $H_1,\ldots,H_n$, in the given order, if the product morphism \[H_1 \times \cdots \times H_n \rightarrow H\] is bijective. For $w \in W$, define $U_w := U \cap w U^- w^{-1}$, where $U$ consists of the unipotent elements of $B$ and similarly, $U^{-}$ is the unipotent part of $B^{-}:=w_0Bw_0$. This is a subgroup of $U$ that is closed and normalized by $T$. Hence, by \cite[\S 14.4]{B91}, $U_w$ is directly spanned, in any order, by the \emph{root subgroups} $U_{\alpha}$, $\alpha \in \Phi^+$, contained in $U_w$. Since by \cite[Part II, 1.4(5)]{J03}, \begin{equation}\label{eqn:conjthat} wU_{\alpha}w^{-1}=U_{w(\alpha)}, \end{equation} these are the $U_{\alpha}$ such that $\alpha \in \Phi^+ \cap w(\Phi^-) = {\mathcal I}(w)$. Thus \begin{equation} \label{eqn:leftinvIdent2.5} U_w = \prod_{\alpha \in {\mathcal I}(w)} U_{\alpha}, \end{equation} where the products $U_{\alpha}$ may be taken in any order. \begin{lemma}\label{prop:anyorder} For a coset $wB \in G/B$, we have \begin{equation} \label{eqn:leftinvIdent3} X_w^{\circ} := BwB = U_w wB = \prod_{\alpha \in {\mathcal I}(w)} U_{\alpha}\,\,wB. \end{equation} Moreover, $X_w^{\circ}$ is isomorphic to the affine space $\mathbb{A}^{\ell(w)}$ (as varieties). \end{lemma} % \begin{proof} % It is a well-known fact that $U_w$ is isomorphic to $X_w^{\circ}$ (as varieties) under the map $u \mapsto uwB,\,u\in U_w$~\cite[\S 14.12]{B91}. The final equality in \eqref{eqn:leftinvIdent3} is \eqref{eqn:leftinvIdent2.5}. By~\cite[Remark in \S 14.4]{B91}, $U_w$ is isomorphic, as a variety, to $\mathbb{A}^{\ell(w)}$. % \end{proof} We say that $w=uv\in W$ is \emph{length additive} if $\ell(uv)=\ell(u)+\ell(v)$. Under this hypothesis, by \cite[Ch. VI, \S 1, Cor. 2 of Prop. 17]{B02} one has \[{\mathcal I}(uv)={\mathcal I}(u)\sqcup u({\mathcal I}(v)).\] Therefore, in particular, if we assume $w_0(I)d \in W$ is \emph{length additive}, then \begin{equation} \label{eqn:splittingUpw0Id} {\mathcal I}(w_0(I)d)={\mathcal I}(w_0(I))\sqcup w_0(I)({\mathcal I}(d)). \end{equation} Define \[V_d := w_0(I) U_d w_0(I)^{-1}=w_0(I)U_d w_0(I).\] \begin{lemma}\label{lemma:iscldsub} $V_d$ is a closed subgroup of $U_{w_0(I)d}$ that is normalized by $T$. \end{lemma} % \begin{proof} % Since $U_d$ is a closed subgroup normalized by $T$, so is $V_d$. Indeed, $V_d$ is a subgroup of $U_{w_0(I)d}$ since % \begin{equation} % \label{eqn:Vddef} % V_d = w_0(I) \prod_{\alpha \in {\mathcal I}(d)} U_{\alpha} w_0(I) = \prod_{\alpha \in w_0(I)({\mathcal I}(d))} U_{\alpha} \leq U_{w_0(I)d}, % \end{equation} % where the first equality is \eqref{eqn:leftinvIdent2.5}, the second is (\ref{eqn:conjthat}), and the subgroup % claim is \eqref{eqn:leftinvIdent2.5} and~\eqref{eqn:splittingUpw0Id}. % \end{proof} \begin{lemma}\label{lemma:directlyspanned} $U_{w_0(I)d}$ is directly spanned by $U_{w_0(I)}$ and $V_d$: \begin{equation} \label{eqn:carefulDirectSpanNotSemidirectProd} U_{w_0(I)d} = U_{w_0(I)} V_d = V_d U_{w_0(I)}. \end{equation} \end{lemma} % \begin{proof} % This follows from \eqref{eqn:leftinvIdent2.5}, \eqref{eqn:splittingUpw0Id}, and \eqref{eqn:Vddef} combined. % \end{proof} Define \[ \tilde{O} := V_d w_0(I)dB \subseteq G/B. \] \begin{lemma} $\tilde{O}$ is $T$-stable for the action $\bigdot$. \end{lemma} % \begin{proof} % The claim follows since % \[V_d w_0(I)dB = (t V_d t^{-1}) tw_0(I)dB \subseteq V_d w_0(I)dB,\] % where the final step follows from the fact that $V_d$ is normalized by $T$ and that $w_0(I)dB$ is a $T$-fixed point in $G / B$. \end{proof} The following is the main point of this section: \begin{proposition} \label{prop:TequivIso} If $w_0(I)d \in W$ is length additive then \[ X_{w_0(I)d}^{\circ} = U_{w_0(I)d}\,w_0(I)dB. \] Hence $\tilde{O} \subset X_{w_0(I)d}^{\circ}$. Moreover, $\tilde{O}$ with the $T$-action $\bigdot$ is $T$-equivariantly isomorphic to $X_{d}^{\circ}$ with the $T$-action $\bigdot_{w_0(I)}$. \end{proposition} % \begin{proof} % By \eqref{eqn:leftinvIdent3}, $X_{w_0(I)d}^{\circ} = U_{w_0(I)d}\,w_0(I)dB$. Combining this with % Lemma~\ref{lemma:iscldsub}, one concludes that $\tilde{O} \subseteq X_{w_0(I)d}^{\circ}$. % Define a map % \begin{alignat*}{3} % \phi: {\tilde O} &\longrightarrow X_{d}^{\circ} \\ % aB &\longmapsto w_0(I)aB. % \end{alignat*} % Now, % \[\phi(\tilde{O}) = w_0(I) V_d\,w_0(I)dB % = U_d\,dB = X_{d}^{\circ},\] % where the second equality is by the definition of $V_d$, and the final equality is Lemma~\ref{prop:anyorder}. Thus $\phi$ is well-defined and surjective. % As $\phi$ is simply left multiplication by $w_0(I)$ it is injective. Further, by Lemma~\ref{prop:anyorder} $X_{d}^{\circ}$ is isomorphic as a variety to $\mathbb{A}^{\ell(d)}$, and thus is smooth, and in particular normal. Hence, by Zariski's main lemma, $\phi$ is an isomorphism of varieties. % To see that $\phi$ is $T$-equivariant for the specified actions, let $t \in T$ and $aB \in \tilde{O}$. Then \[ \phi(t \bigdot aB) = w_0(I)taB = w_0(I)tw_0(I)w_0(I)aB = \gamma_{w_0(I)}(t) \bigdot \phi(aB) = t \bigdot_{w_0(I)} \phi(aB).\qedhere\]\end{proof} \section{Proof of the main result}\label{sec:3} We need a lemma examining the $L_I$-action on $\tilde{O}$. This lemma is then used in conjunction with Proposition~\ref{prop:TequivIso} to prove our main result. Let $B_{L_I}=L_I\cap B$ and let $U_{L_I}$ be the unipotent radical of $B_{L_I}$. Then $B_{L_I}$ is a Borel subgroup in $L_I$~\cite[\S 14.17]{B91} with $U_{L_I}=B_{L_I}\cap U$ and $B_{L_I}=T \ltimes U_{L_I}$. Since $L_I$ is the subgroup of $G$ generated by $T$ and $\{U_{\alpha} \,\mid\, \alpha \in \Phi(I)\}$~\cite[\S 3.2.2]{LR08}, it is straightforward to show that \[U_{L_I} = \prod_{\alpha \in \Phi^+(I)} U_{\alpha},\] where the product is taken in any order~\cite[\S 14.4]{B91}. \begin{lemma} \label{lemma:orbitDynamics} Let $w = w_0(I) d \in W$ be length additive. Let $x \in X_{w_0(I)d}^{\circ} \setminus \tilde{O}$ and $y,z \in \tilde{O}$. \begin{itemize} \item[(i)] $uy \notin \tilde{O}$ for all $u \in U_{L_I}$ with $u \neq e$. \item[(ii)] $tx \notin \tilde{O}$ for all $t \in T$. \item[(iii)] There exists $b \in B_{L_I}$ such that $by=z$ if and only if there exists $t \in T$ such that $ty = z$. \end{itemize} \end{lemma} % \begin{proof} % (i) We have \[ U_{L_I} = \prod_{\alpha \in \Phi^+(I)} U_{\alpha} = U_{w_0(I)}, \] where the final equality is \eqref{eqn:leftinvIdent2.5}. Thus $u \in U_{w_0(I)}$. % Since $y \in \tilde{O}$, we have that $y=v w_0(I)dB$ for some $v \in V_d$. By % Lemma~\ref{lemma:directlyspanned}, $uv \in U_{w_0(I)d} \setminus V_d$ for $u \neq e$. Thus $uv w_0(I)dB \in X_{w_0(I)d}^{\circ} \setminus \tilde{O}$ by Lemma~\ref{prop:anyorder}. % \noindent (ii) This follows immediately from the fact that $\tilde{O}$ is $T$-stable. % \noindent (iii) The Borel $B_{L_I} = T \ltimes U_{L_I}$, and thus for all $b \in B_{L_I}$ we may express $b=tu$ for unique $t\in T, u \in U_{L_I}$. If $u \neq e$, then $uy \notin \tilde{O}$ by (i) and so $by=tuy\notin \tilde{O}$ by (ii). Hence, if $by=z$, then $u = e$ and $b = t \in T$. The converse direction is immediate since $T \subseteq B_{L_I}$. % \end{proof} We now have the necessary ingredients to complete the proof of Theorem~\ref{conj:main}. % \noindent \emph{Proof of Theorem~\ref{conj:main}:} ($\Leftarrow$) Let $w$ be $I$-spherical. Then $w = w_0(I) c$ is length additive and $c$ is a standard Coxeter element. Our goal is to exhibit a $x\!\in\!\tilde{O}$ such that $\dim(B_{L_I}\!\bigdot\!x)\!=\!\dim X_{w_0(I)c}^{\circ}$. % The Schubert variety $X_{c}$ is a toric variety~\cite{Karrup}; it contains an open, dense $T$-orbit $\mathcal{O}$ for the $T$-action $\bigdot$. Since $X_{c}^{\circ}$ is an open, dense subset of $X_{c}$, $\mathcal{O} \cap X_{c}^{\circ}$ is open and dense in $X_{c}^{\circ}$; since $X_{c}^{\circ}$ is $T$-stable we have that $\mathcal{O} \cap X_{c}^{\circ}$ is a $T$-orbit in $X_{c}^{\circ}$ for the $T$-action $\bigdot$. Lemma~\ref{lemma:denseByAutomorphism} implies that $\mathcal{O} \cap X_{c}^{\circ}$ is an open, dense $T$-orbit for the $T$-action $\bigdot_{w_0(I)}$. % By Proposition~\ref{prop:TequivIso}, there is a $T$-equivariant isomorphism $\phi: % \tilde{O}\to X_{c}^{\circ}$. Let % \[\Theta=\phi^{-1}({\mathcal O}\cap X_c^{\circ});\] % this is an open, dense $T$-orbit in $\tilde{O}$ for the $T$-action $\bigdot$. Let $x \in \Theta$. By Lemma~\ref{lemma:orbitDynamics}(iii), the isotropy subgroup $(B_{L_I})_x$ is equal to the isotropy subgroup $T_x$. By \cite[Proposition 1.11]{B09}, for any variety $X$ equipped with the action of an algebraic group $H$, the orbit $H \cdot x$, $x \in X$, is a subvariety of $X$ of dimension $\dim H - \dim H_x$, % \begin{equation} % \label{eqn:orbitDim} % \dim(H \cdot x) = \dim H - \dim H_x. % \end{equation} % The above combine to imply that % \begin{equation} % \label{eqn:stabDim} % \dim(B_{L_I})_x = \dim T_x = \dim T - \dim(T \bigdot x) = \dim T - \dim \Theta = \dim T - \ell(c). % \end{equation} % We conclude, applying \eqref{eqn:orbitDim} and \eqref{eqn:stabDim}, that % \begin{align*} % \dim(B_{L_I} \cdot x) & = \dim B_{L_I} - \dim (B_{L_I})_x\\ % & = \ell(w_0(I)) + \dim T - (\dim T - \ell(c))\\ % & = \ell(w_0(I)) + \ell(c) \\ % & = \ell(w_0(I)c), % \end{align*} % and thus there exists an dense $B_{L_I}$-orbit in $X_{w_0(I)c}$. Indeed, this dense orbit must also be open in its closure by~\cite[Proposition 1.8]{B91}. Hence, $X_{w_0(I)c}$ is $L_I$-spherical. % \noindent($\Rightarrow$) Suppose $w$ is not $I$-spherical. Then $w = w_0(I) d$ where $d$ is not a standard % Coxeter element. Moreover, by the hypothesis that $I\subseteq {\mathcal D}_L(w)$, this factorization % is length additive. % The Schubert variety $X_{d}$ is not a toric variety for the $\bigdot$ action of $T$~\cite{Karrup}. If $X_{d}^{\circ}$ contained an open, dense $T$-orbit, then $X_{d}$ would be a toric variety for $\bigdot$. Thus $X_{d}^{\circ}$ is not a toric variety for $\bigdot$. In general, a normal $G$-variety is spherical if and only if it has finitely many $B$-orbits % (see \cite[Theorem~2.1.2]{Perrin}). If $G=T$ then $B=T$ and hence there are infinitely many $T$-orbits in $X_{d}^{\circ}$ for $T$-action $\bigdot$; and for the $T$-action $\bigdot_{w_0(I)}$ by Lemma~\ref{lemma:denseByAutomorphism}. % By Proposition~\ref{prop:TequivIso}, % $\tilde{O}$ % is $T$-equivariantly isomorphic as an affine variety to $X_{d}^{\circ}$. Thus, there are infinitely many $T$-orbits in $\tilde{O}$ for $T$-action $\bigdot$. Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two such orbits, and $x_1 \in \mathcal{O}_1, x_2 \in \mathcal{O}_2$. The fact that $x_1$ and $x_2$ reside in different orbits implies that there does not exist a $t \in T$ such that $tx_1 = x_2$. Thus Lemma~\ref{lemma:orbitDynamics}(iii) implies % $B_{L_I} \bigdot x_1 \cap B_{L_I} \bigdot x_2 = \emptyset$. % As these were an arbitrary pair among the infinite $T$-orbits, there must be infinitely many $B_{L_I}$ orbits in $X_{w_0(I)d}^{\circ}$ and hence in $X_{w_0(I)d}$. We conclude that $X_{w_0(I)d}$ is not $L_I$-spherical by % the same result \cite[Theorem~2.1.2]{Perrin} mentioned above. % \qed \section{Application to Demazure modules}\label{sec:4} As an application of these results we give a sufficient condition for a Demazure module to be a multiplicity-free $L_I$-module; equivalently, a sufficient condition for a Demazure character to be multiplicity-free with respect to the basis of irreducible $L_I$-characters. Let $\mathfrak{X}(T)$ denote the lattice of weights of $T$; our fixed Borel subgroup $B$ determines a subset of dominant integral weights $\mathfrak{X}(T)^+ \subset \mathfrak{X}(T)$. The finite-dimensional irreducible $G$-representations are indexed by $\lambda \in \mathfrak{X}(T)^+$. Denoting the associated representation by $V_{\lambda}$, there is a class of $B$-submodules of $V_{\lambda}$, first introduced by Demazure \cite{Demazure}, that are indexed by $w \in W$. If $v_{\lambda}$ is a nonzero highest weight vector, then the \emph{Demazure module} $V_{\lambda}^{w}$ is the minimal $B$-submodule of $V_{\lambda}$ containing $wv_{\lambda}$. There is a geometric construction of these Demazure modules. For $\lambda \in \mathfrak{X}(T)^+$, let $\mathfrak{L}_{\lambda}$ be the associated line bundle on $G / B$. For $w \in W$, we write $\mathfrak{L}_{\lambda}|_{X_w}$ for the restriction of $\mathfrak{L}_{\lambda}$ to the Schubert subvariety $X_w \subseteq G/ B$. Then the Demazure module $V_{\lambda}^{w}$ is isomorphic to the dual of the space of global sections of $\mathfrak{L}_{\lambda}|_{X_w}$, that is \[V_{\lambda}^{w} \cong H^0(X_w, \mathfrak{L}_{\lambda}|_{X_w})^*.\] This geometric perspective highlights the fact that $V_{\lambda}^{w}$ is not just a $B$-module, but is in fact also a $L_I$-module via the action induced on $H^0(X_w, \mathfrak{L}_{\lambda}|_{X_w})$ by the left multiplication action of $L_I$ on $X_w$. As $L_I$ is a reductive group over characteristic zero, any $L_I$-module decomposes into a direct sum of irreducible $L_I$-modules. Let $\mathfrak{X}_{L_I}(T)^+$ be the set of dominant integral weights with respect to the choice of maximal torus and Borel subgroup $T \subseteq B_I \subseteq L_I$. For $\mu \in \mathfrak{X}_{L_I}(T)^+$, let $V_{L_I,\mu}$ be the associated irreducible $L_I$-module. If $M$ is a $L_I$-module and \[ \displaystyle M = \bigoplus_{\mu \in \mathfrak{X}_{L_I}(T)^+} V_{L_I,\mu}^{\oplus m_{L_I,\mu}} \] is the decomposition into irreducible $L_I$-modules, then we say that $M$ is a \emph{multiplicity-free $L_I$-module} if $m_{L_I,\mu} \in \{ 0, 1 \}$. Similarly, if $\mathrm{char}(M)$ is the formal $T$-character of $M$ and \[ \displaystyle \mathrm{char}(M) = \sum_{\mu \in \mathfrak{X}_{L_I}(T)^+} m_{L_I,\mu} \mathrm{char}(V_{L_I,\mu}), \] then we say that $\mathrm{char}(M)$ is \emph{$I$-multiplicity-free} if $m_{L_I,\mu} \in \{ 0, 1 \}$. The following argument was given for type $A$ in \cite[Theorem~4.13(II)]{Hodges.Yong1}. We prove the general type argument (which is essentially the same) for sake of completeness: \begin{theorem} \label{theorem:demazureConsequences} Let $w \in W$ with $I \subseteq D_L(w)$. Then $X_w$ is $L_I$-spherical if and only if for all $\lambda \in \mathfrak{X}(T)^+$, the Demazure module $V_{\lambda}^{w}$ is multiplicity-free $L_I$-module. \end{theorem} % \begin{proof} % Let $X$ be a quasi-projective, normal variety with the action of a complex, connected, reductive algebraic group $G$. Then $X$ is $G$-spherical if and only if the $G$-module $H^0(X,\mathfrak{L})$ is a multiplicity free $G$-module for all $G$-linearized line bundles $\mathfrak{L}$~\cite[Theorem 2.1.2]{Perrin}. % All Schubert varieties $X_w \subseteq G / B$ are normal, quasi-projective varieties~\cite{Jos.Normal}. Further, as $L_I$ is reductive and we are in characteristic zero, $V_{\lambda}^{w}$ is a multiplicity-free $L_I$-module if and only if its dual space $(V_{\lambda}^{w})^* = H^0(X_w, \mathfrak{L}_{\lambda}|_{X_w})$ is a multiplicity-free $L_I$-module~\cite[\S 31.6]{Humphreys}. The above combines to imply our desired result once we show that the $L_I$-linearized line bundles on $X_w$ are precisely of the form $\mathfrak{L}_{\lambda}|_{X_w}$ for $\lambda \in \mathfrak{X}(T)^+$. % The line bundles, with non-trivial spaces of global sections, on $G/B$ are precisely $\mathfrak{L}_{\lambda}$, for $\lambda \in \mathfrak{X}(T)^+$; % these line bundles are all $G$-linearized~\cite[\S 1.4]{Brion.Lectures}. Every line bundle on $X_w$ is the restriction of a line bundle on $G/B$~\cite[Proposition 2.2.8]{Brion.Lectures}. We are done since the restriction $\mathfrak{L}_{\lambda}|_{X_w}$ of the $G$-linearized line bundle $\mathfrak{L}_{\lambda}$, for $\lambda \in \mathfrak{X}(T)^+$, is $L_I$-linearized. % \end{proof} \begin{corollary} Let $w \in W$ be $I$-spherical for $I \subseteq D_L(w)$. For all $\lambda \in \mathfrak{X}(T)^+$, the Demazure module $V_{\lambda}^{w}$ is a multiplicity-free $L_I$-module. \end{corollary} % \begin{proof} % By Theorem~\ref{conj:main}, if $w$ is $I$-spherical then $X_w$ is $L_I$-spherical. Therefore, by Theorem~\ref{theorem:demazureConsequences}, $V_{\lambda}^{w}$ is a multiplicity-free $L_I$-module for $\lambda \in \mathfrak{X}(T)^+$. % \end{proof} \begin{corollary} Let $w \in W$ be $I$-spherical for $I \subseteq D_L(w)$. For all $\lambda \in \mathfrak{X}(T)^+$, the Demazure character $\mathrm{char}(V_{\lambda}^{w})$ is $I$-multiplicity-free. \end{corollary} These two corollaries appear non-trivial from a combinatorial perspective, even for a \emph{specific choice} of dominant weight $\lambda$ with fixed $w\in W$. The Demazure character can be recursively computed using Demazure operators. There is also a combinatorial rule for the character in terms of crystal bases (in instantiations such as the \emph{Littelmann path model} or the \emph{alcove walk model}); see, e.g., the textbook \cite{Bump}. However, an argument based on these methods eludes in general type, although we have an argument in type $A$ \cite{GHY}. %\acknowledgements{RH was partially supported by an AMS-Simons Travel Grant. AY is partially supported by a Simons Collaboration Grant and an NSF RTG in Combinatorics.} %% if you use biblatex then this generates the bibliography %% if you use some other method then remove this and do it your own way \printbibliography %\bibliographystyle{plain} %\bibliography{GHY.FPSAC2024} \end{document}