A proof of the refined PRV conjecture via the cyclic convolution variety

TL;DR

This paper uses geometric Satake correspondence and cyclic convolution varieties to provide a simple proof of the PRV conjecture and its refinement, connecting MV-cycles with orbit closures.

Contribution

It offers a novel, geometric proof of the PRV conjecture and its refinement using the cyclic convolution variety and MV-cycles, clarifying their orbit structure.

Findings

Proof of the PRV conjecture using geometric methods

Refinement of the PRV conjecture established

Explicit characterization of MV-cycles as orbit closures

Abstract

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar's refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

Equations

$${\left(V(\lambda )\otimes V(\mu )\otimes V(\nu )\right)}^{\check{G}}\ne (0).$$

$$\dim {\left(V(\lambda )\otimes V(\mu )\otimes V(\nu )\right)}^{\check{G}}\ge m_{\lambda ,\mu ,w}.$$

$$(L_1,L_2)=g([0],[\lambda ])$$

$${\mathrm{Gr}}_{G,c(\vec{\lambda })}:=\left\{(L_1,\dots ,L_s)\in {\left(G(\mathcal{K})/G(\mathcal{O})\right)}^s\mid L_s=[0],d(L_{i-1},L_i)=[{\lambda }_i]\forall i\right\},$$

$$\dim {\left(V({\lambda }_1)\otimes \cdots \otimes V({\lambda }_s)\right)}^{\check{G}};$$

$$[{\Omega }_{w^{-1}}]{\odot }_0[{\Omega }_e]{\odot }_0[X_{w^{-1}}]=1,$$

$$([0],[\lambda ])$$

$$([\lambda ],[\lambda +w\mu ])$$

$$([\lambda +w\mu ],[0])$$

$$G(\mathcal{O})\to G(\mathcal{O}/(t^N)).$$

$${\mathrm{Stab}}_{G(\mathcal{O})}(x)=G(\mathcal{O})\cap t^{\lambda }G(\mathcal{O})t^{-\lambda }\cap t^{\lambda +w\mu }G(\mathcal{O})t^{-\lambda -w\mu }\subseteq G(\mathcal{O})$$

$$\mathfrak{g}(\mathcal{O})\cap {\mathrm{Ad}}_{t^{\lambda }}\mathfrak{g}(\mathcal{O})\cap {\mathrm{Ad}}_{t^{\lambda +w\mu }}\mathfrak{g}(\mathcal{O})\simeq \mathfrak{h}(\mathcal{O})\oplus \underset{\alpha \in \Phi }{⨁}{t}^{\max (0,⟨\alpha ,\lambda ⟩,⟨\alpha ,\lambda +w\mu ⟩)}{\mathfrak{g}}_{\alpha }(\mathcal{O})$$

$$\mathrm{Lie}({\mathrm{Stab}}_{G(\mathcal{O}/(t^N))}(x))\simeq \mathfrak{h}(\mathcal{O}/(t^N))\oplus \underset{\alpha \in \Phi }{⨁}{t}^{\max (0,⟨\alpha ,\lambda ⟩,⟨\alpha ,\lambda +w\mu ⟩)}{\mathfrak{g}}_{\alpha }(\mathcal{O}/(t^N));$$

$$\sum _{\alpha \in \Phi }\max (0,⟨\alpha ,\lambda ⟩,⟨\alpha ,\lambda +w\mu ⟩).$$

$$\sum_{\alpha\succeq 0}\max(0,\langle\alpha,\lambda\rangle,\langle\alpha,\lambda+w\mu\rangle)=\sum_{\alpha\succeq 0}\langle\alpha,\lambda\rangle+\sum_{\begin{array}[]{c}\alpha\succeq 0\\ \langle\alpha,w\mu\rangle\geq 0\end{array}}\langle\alpha,w\mu\rangle.$$

$$\sum_{\begin{array}[]{c}\alpha\succeq 0\\ w^{-1}\alpha\succeq 0\end{array}}\langle\alpha,w\mu\rangle=\sum_{\Phi^{+}\cap w\Phi^{+}}\langle\alpha,w\mu\rangle,$$

$$\sum_{\begin{array}[]{c}\alpha\preceq 0\\ \langle\alpha,-v^{-1}\nu\rangle\geq 0\end{array}}\langle\alpha,-v^{-1}\nu\rangle=\sum_{\begin{array}[]{c}\alpha\succeq 0\\ \langle\alpha,v^{-1}\nu\rangle\geq 0\end{array}}\langle\alpha,v^{-1}\nu\rangle.$$

$$⟨2\rho ,\lambda ⟩+⟨\rho +w\rho ,$$

$$=⟨\rho ,\lambda +w\mu +v^{-1}\nu ⟩+⟨\rho ,\lambda ⟩+⟨w\rho ,w\mu ⟩+⟨v^{-1}\rho ,v^{-1}\nu ⟩$$

$$=0+⟨\rho ,\lambda +\mu +\nu ⟩.$$

$$([0],[\lambda ])$$

$$([\lambda ],[\lambda +u\mu ])$$

$$([\lambda +u\mu ],[0])$$

$$e{v}_0:G{r}_{\lambda }\to G/P_{\lambda },$$

$$g{q}^{\prime -1}{w}_0^{-1}[-w_0\nu ]=q^{-1}{w}_0^{-1}[-w_0\nu ],$$

$$P_{\lambda }{q}^{\prime -1}{w}_0^{-1}{P}_{-w_0\nu }=P_{\lambda }{q}^{-1}{w}_0^{-1}{P}_{-w_0\nu }$$

$$W_{\lambda }{q}^{\prime -1}{w}_0^{-1}{W}_{-w_0\nu }=W_{\lambda }{q}^{-1}{w}_0^{-1}{W}_{-w_0\nu }$$

$$\lambda +u\mu =q^{-1}{w}_0^{-1}(-w_0\nu )=r{q}^{\prime -1}{w}_0^{-1}(-w_0\nu )=r(\lambda +u^{\prime }\mu )=\lambda +r{u}^{\prime }\mu ;$$

$$\pm {\alpha }^{\vee }=-{\alpha }^{\vee }\pm {\alpha }^{\vee }$$

$$([0],[{\alpha }^{\vee }])$$

$$([{\alpha }^{\vee }],\bar{y})$$

$$(\bar{y},[0])$$

$$y=\left[\begin{array}[]{cc}t&1\\ 0&t^{-1}\end{array}\right]=\left[\begin{array}[]{cc}t&0\\ 0&t^{-1}\end{array}\right]\left[\begin{array}[]{cc}0&1\\ -1&t\end{array}\right]\left[\begin{array}[]{cc}t&0\\ 0&t^{-1}\end{array}\right]\left[\begin{array}[]{cc}1&0\\ t&1\end{array}\right]$$

$$\left[\begin{array}[]{cc}t&0\\ 0&t^{-1}\end{array}\right]\left[\begin{array}[]{cc}0&1\\ -1&t\end{array}\right]\left[\begin{array}[]{cc}t&0\\ 0&t^{-1}\end{array}\right]\left[\begin{array}[]{cc}0&1\\ -1&t\end{array}\right]\left[\begin{array}[]{cc}t&0\\ 0&t^{-1}\end{array}\right]=\left[\begin{array}[]{cc}-t&1\\ -1&0\end{array}\right].$$

Full text

1. Introduction

We give a short proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture first proven independently by Kumar in [K] and Mathieu in [M]. Our method extends to give a proof of Verma’s refined conjecture which was first proven by Kumar in [Kr].

Let $\check{G}$ be a complex reductive group, whose representation theory we are interested in. (We reserve the symbol $G$ for the complex reductive Langlands dual group of $\check{G}$ because $G$ will be used more prominently in the proof, which goes through the geometric Satake correspondence.) Fix a maximal torus $\check{T}$ and Borel subgroup $\check{B}$ of $\check{G}$. Let $W$ be the Weyl group of $\check{G}$ (equivalently, of $G$). The statement of the original theorem is

Theorem 1.1** (PRV conjecture).**

Let $\lambda,\mu$ be dominant weights for $\check{G}$ with respect to $\check{B}$, and let $w\in W$ be any Weyl group element. Find $v\in W$ so that $\nu:=v(-\lambda-w\mu)$ is dominant. Then

[TABLE]

Kumar proved a refinement of this theorem in [Kr] regarding the dimensions of the spaces of invariants. Let $W_{\delta}$ for any weight $\delta$ denote the stabilizer subgroup of $\delta$ in $W$. The stronger theorem is

Theorem 1.2** (Refinement).**

Let $\lambda,\mu,\nu,w$ be as above. Let $m_{\lambda,\mu,w}$ count the number of distinct cosets $\bar{u}\in W_{\lambda}\backslash W/W_{\mu}$ such that $-\lambda-w\mu$ and $-\lambda-u\mu$ are $W$-conjugate (equivalently, $\nu$ can be written $q(-\lambda-u\mu)$ for some $q\in W$). Then

[TABLE]

In particular, since $m_{\lambda,\mu,w}\geq 1$ by definition, the second theorem implies the first.

We will use properties of a certain complex variety called the cyclic convolution variety, whose definition we recall; see [H]§2, although our symmetric formulation is from [Kam]§1. Let $G$ be the Langlands dual group to $\check{G}$ with dual torus $T$ and Borel subgroup $B$. Let $\lambda_{i},i=1,\ldots,s$ be a collection of dominant weights for $\check{G}$ w.r.t. $\check{B}$; these induce dominant coweights of $G$ w.r.t. $B$. Set $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}=\mathbb{C}[[t]]$. Each cocharacter $\lambda:\mathbb{C}^{\times}\to T$ induces an element $t^{\lambda}$ of $G(\mathcal{K})$; denote by $[\lambda]$ its image in $G(\mathcal{K})/G(\mathcal{O})$. Recall that via the Chevalley decomposition any two points $L_{1},L_{2}$ in $G(\mathcal{K})/G(\mathcal{O})$ give rise to a unique dominant coweight $\lambda$ of $T$ such that

[TABLE]

for some $g\in G(\mathcal{K})$; we write $\lambda=d(L_{1},L_{2})$ to convey this information concisely.

The cyclic convolution variety is

[TABLE]

where we take $L_{0}$ to mean $L_{s}$. The maximum possible dimension of $\operatorname{Gr}_{G,c(\vec{\lambda})}$ is $\langle\rho,\sum\lambda_{i}\rangle$, where $\rho$ is the usual half-sum of positive roots for $G$, and via the geometric Satake correspondence ([L, G, BD, MV]) the number of irreducible components of this dimension (if any) is equal to

[TABLE]

see also [H]*Proposition 3.1.

Our task is therefore to produce irreducible components of the right dimension, which we find as $G(\mathcal{O})$-orbit closures of suitable points. These are in bijection with certain MV-cycles which we make explicit. As a corollary we obtain the following known result:

Corollary 1.3**.**

Let $\lambda,\mu$ be dominant and $w\in W$ such that $\nu:=\lambda+w\mu$ is also dominant. Then the multiplicity of $V(\nu)$ inside $V(\lambda)\otimes V(\mu)$ is exactly $1$.

This is already known from a multiplicity theorem of Kostant; see [Ko]*Lemma 4.1 and [Kumon]*Corollary 3.8. (It is also a consequence of Roth’s theorem [Roth] where $P_{I}=B$, $\bar{G}=\{1\}$, and the Schubert calculus equation is

[TABLE]

using the notation found there.)

Our technique of producing components of the right dimension is not limited to the PRV setting; we illustrate this by an explicit example in Section 5.

Our proof of Theorem 1.1 should be compared with the proof of [Rz]*Lemma 5.5, where a geometric analogue of PRV is proved. There a one-sided dimension estimate on a fibre of the convolution morphism provides existence of components of the correct dimension, but the fibre component is not realized as the (closure of an) orbit under a group action; nor is the specific MV-cycle mentioned. The lower bound on number of components (yielding the refined version) is not made there.

See also [H2]*Theorem 6.1, where non-emptiness of the relevant variety (but not its dimension) is established, implying Theorem 1.1 only in the case where $\lambda,\mu$ are sums of minuscule coweights.

In an appendix, joint with P. Belkale, we describe the relationship of this work to a more general question on the transfer of invariants between Langlands dual groups, with the PRV case corresponding to the inclusion of a maximal torus inside a reductive group.

1.1. Acknowledgements

I thank Shrawan Kumar, Prakash Belkale, Joel Kamnitzer, and Marc Besson for helpful discussions and suggestions.

2. Proof of the conjecture

We will need some additional notation: let $\Phi$ denote the set of roots of $G$, and for $\alpha\in\Phi$ let $\alpha\succeq 0$ mean $\alpha$ is a positive root w.r.t. $B$ (likewise $\alpha\preceq 0$ means $-\alpha\succeq 0$).

Proof.

Step 1 We claim that the cyclic convolution variety $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ is nonempty. Indeed, the point $x=([\lambda],[\lambda+w\mu],[0])$ satisfies

[TABLE]

Step 2 Observe that any $\operatorname{Gr}_{G,c(\vec{\lambda})}$ has a $G(\mathcal{O})$-diagonal action on the left. We claim that the orbit $G(\mathcal{O})x\subseteq\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ is a finite-dimensional subvariety and has dimension $\langle\rho,\lambda+\mu+\nu\rangle$; this will conclude the proof, since the connectedness of $G(\mathcal{O})$ means $G(\mathcal{O})x$ is contained in an irreducible component of $\operatorname{Gr}_{G,c(\vec{\lambda})}$ necessarily of dimension $\langle\rho,\lambda+\mu+\nu\rangle$.

For any integer $N>0$, let $K_{N}$ denote the kernel of the surjective group homomorphism

[TABLE]

Observe that, for high enough $N\gg 0$, $K_{N}$ stabilizes the point $x$ (it suffices to embed $G$ into some $GL_{m}$ and examine matrix entries). Therefore $G(\mathcal{O})x$ has a transitive action by the finite-dimensional linear algebraic group $G(\mathcal{O}/(t^{N}))$.

The stabilizer $\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x)$ is the image of

[TABLE]

under the quotient; i.e., $\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x)=\operatorname{Stab}_{G(\mathcal{O})}(x)/K_{N}$.

By the orbit-stabilizer theorem, $G(\mathcal{O}/(t^{N}))x\simeq G(\mathcal{O}/(t^{N}))/\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x)$. As

$G(\mathcal{O}/(t^{N}))/\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x)$ is a smooth finite-dimensional variety, we may calculate its dimension by the dimension of its tangent space at the origin. For an arbitrary group scheme $H$ over $\mathbb{C}$, one takes $\operatorname{Lie}(H)$ to mean the kernel of $H(\mathbb{C}[\epsilon]/(\epsilon^{2}))\xrightarrow{\epsilon\mapsto 0}H(\mathbb{C})$. Since Lie commutes with intersections (of subgroups of $G(\mathcal{K})$, see [Milne]*§10.c), $\operatorname{Lie}(\operatorname{Stab}_{G(\mathcal{O})}(x))$ is

[TABLE]

Thus in the quotient

[TABLE]

note that, for every $\alpha$, $\langle 0,\alpha\rangle\leq N$ and $\langle\alpha,\lambda+w\mu\rangle\leq N$ so that $K_{N}\subseteq\operatorname{Stab}_{G(\mathcal{O})}(x)$. For the finite-dimensional affine group scheme $S:=\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x)$, $\operatorname{Lie}(S)$ is naturally identified with the tangent space of $S$ at the identity. Therefore the $\mathbb{C}$-dimension of the tangent space $\mathfrak{g}(\mathcal{O}/(t^{N}))/\operatorname{Lie}(\operatorname{Stab}_{G(\mathcal{O}/(t^{N}))}(x))$ is

[TABLE]

The proof of the claim therefore reduces to the following calculation.

Step 3 We claim that $\langle\rho,\lambda+\mu+\nu\rangle=\sum_{\alpha\in\Phi}\max(0,\langle\alpha,\lambda\rangle,\langle\alpha,\lambda+w\mu\rangle).$ Let us examine the sum on the right in two parts, summing over $\alpha\preceq 0$ and $\alpha\succeq 0$ separately.

If $\alpha\succeq 0$, then $\max(0,\langle\alpha,\lambda\rangle,\langle\alpha,\lambda+w\mu\rangle)=\max(\langle\alpha,\lambda\rangle,\langle\alpha,\lambda+w\mu\rangle)$ due to the dominance of $\lambda$. Furthermore, $\langle\alpha,\lambda\rangle$ will be the bigger of the two unless $\langle\alpha,w\mu\rangle\geq 0$. Therefore

[TABLE]

The first sum on the RHS is clearly equal to $\langle 2\rho,\lambda\rangle$. As for the second sum, observe that $\langle\alpha,w\mu\rangle\geq 0\iff\langle w^{-1}\alpha,\mu\rangle\geq 0$. As $\mu$ is dominant, this happens only when $w^{-1}\alpha\succeq 0$ or when $w^{-1}\alpha\preceq 0$ and $\langle w^{-1}\alpha,\mu\rangle=0$. The latter class of $\alpha$ doesn’t contribute to the sum, so that second RHS term is equal to

[TABLE]

where $\Phi^{+}$ denotes the set of positive roots. As is well known (see for example [Ku2]*1.3.22.3), $\sum_{\Phi^{+}\cap w\Phi^{+}}\alpha=\rho+w\rho.$ Putting everything together so far, the original sum over $\alpha\succeq 0$ yields $\langle 2\rho,\lambda\rangle+\langle\rho+w\rho,w\mu\rangle$.

If $\alpha\preceq 0$, then $\max(0,\langle\alpha,\lambda\rangle,\langle\alpha,\lambda+w\mu\rangle)=\max(0,\langle\alpha,\lambda+w\mu\rangle)$. Recall that $\lambda+w\mu=-v^{-1}\nu$; therefore the sum over $\alpha\preceq 0$ is

[TABLE]

As before, this equals $\langle\rho+v^{-1}\rho,v^{-1}\nu\rangle$. Finally, we conclude as desired that the dimension of the space in question is

[TABLE]

∎

3. Proof of the refinement

Proof.

Suppose $u\in W$ is such that $\nu=q(-\lambda-u\mu)$ for some $q\in W$. Then $x(u):=([\lambda],[\lambda+u\mu],[0])$ satisfies

[TABLE]

therefore $x(u)\in\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ and $G(\mathcal{O})x(u)$ is a subvariety of $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ of dimension $\langle\rho,\lambda+\mu+\nu\rangle$ for exactly the same reason as before.

Claim If $x(u)=gx(u^{\prime})$ for some $g\in G(\mathcal{O})$, then $\bar{u}=\bar{u}^{\prime}\in W_{\lambda}\backslash W/W_{\mu}$.

Proof Assume $x(u)=gx(u^{\prime})$ for some $g\in G(\mathcal{O})$. Fix $q,q^{\prime}$ satisfying $\nu=q(-\lambda-u\mu)=q^{\prime}(-\lambda-u^{\prime}\mu)$. We are given that $g[\lambda]=[\lambda]$ and $g[\lambda+u^{\prime}\mu]=[\lambda+u\mu]$. First we demonstrate that we can replace $g$ with an element of $G$. Recall from [MV] that there is a map

[TABLE]

where $P_{\lambda}$ is the smallest parabolic containing $B^{-}$ and $L_{\lambda}$, where $B^{-}$ the Borel opposite to $B$ and $L_{\lambda}$ is the centralizer of $t^{\lambda}$ in $G$. The map is given by $g(t)t^{\lambda}G(\mathcal{O})\mapsto g(0)P_{\lambda}$ and makes $Gr_{\lambda}$ an affine bundle over $G/P_{\lambda}$.

We find that $g(0)P_{\lambda}=P_{\lambda}$ by taking $ev_{0}$ of both sides of the equation $g[\lambda]=[\lambda]$; i.e., $g(0)\in P_{\lambda}$. From $\nu=q(-\lambda-u\mu)$ we have $-w_{0}\nu=w_{0}q(\lambda+u\mu)$. The second equation can be formulated as

[TABLE]

which under $ev_{0}$ gives $g(0)q^{\prime-1}w_{0}^{-1}P_{-w_{0}\nu}=q^{-1}w_{0}^{-1}P_{-w_{0}\nu}.$

We now attempt to replace $g(0)$ with a Weyl group element, as follows. Since $g(0)\in P_{\lambda}$, the double cosets

[TABLE]

agree, in which case

[TABLE]

by [BT]*Corollaire 5.20 (see also [Kr]*Lemma 2.2). Writing $rq^{\prime-1}w_{0}^{-1}r^{\prime}=q^{-1}w_{0}^{-1}$ for some $r\in W_{\lambda},r^{\prime}\in W_{-w_{0}\nu}$, observe that

[TABLE]

therefore $ru^{\prime}\mu=u\mu$ and thus $ru^{\prime}W_{\mu}=uW_{\mu}$. This gives $W_{\lambda}u^{\prime}W_{\mu}=W_{\lambda}uW_{\mu}$ as desired.

So for any pair $\bar{u},\bar{u}^{\prime}$ distinct in $W_{\lambda}\backslash W/W_{\mu}$ (such that $-\lambda-u\mu$ and $-\lambda-u^{\prime}\mu$ are both conjugate to $\nu$), the orbits $G(\mathcal{O})x(u)$ and $G(\mathcal{O})x(u^{\prime})$ must be disjoint. Each orbit $G(\mathcal{O})x(u)$ is irreducible, so the closure $\overline{G(\mathcal{O})x(u)}$ inside $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ is an irreducible component of the same (top) dimension. Disjoint orbits necessarily give distinct (possibly not disjoint) irreducible components. Therefore the number of irreducible components of the top dimension of $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ is at least $m_{\lambda,\mu,w}$, from which the theorem follows. ∎

4. Relation to MV-cycles

Here we recall the summary of the geometric Satake correspondence as presented in [A]. Let $\mathcal{F}_{\nu}=\pi^{-1}([\nu])$ be the fibre of the natural projection map

{\overline{\operatorname{Gr}_{\lambda}}\tilde{\times}\overline{\operatorname{Gr}_{\mu}}:=\{(aG(\mathcal{O}),bG(\mathcal{O}))\in\overline{\operatorname{Gr}_{\lambda}}\times\overline{\operatorname{Gr}_{\lambda+\mu}}\mid a^{-1}bG(\mathcal{O})\in\overline{\operatorname{Gr}_{\mu}}\}}$${\overline{\operatorname{Gr}_{\lambda+\mu}}}$$\scriptstyle{\pi}

over $[\nu]$, where $\nu\preceq\lambda+\mu$. Then the multiplicity of $V(\nu)$ inside $V(\lambda)\otimes V(\mu)$ is equal to the number of irreducible components of $\mathcal{F}_{\nu}$ of dimension $\langle\rho,\lambda+\mu-\nu\rangle$, the maximal possible dimension. (There is a $1-1$ correspondence between these irreducible components and those of top dimension in $\operatorname{Gr}_{G,c(\lambda,\mu,-w_{0}\nu)}$.) According to [A]*Theorem 8, the irreducible components of $\mathcal{F}_{\nu}$ of dimension $\langle\rho,\lambda+\mu-\nu\rangle$ are exactly the Mirković-Vilonen cycles for $\overline{\operatorname{Gr}_{\lambda}}$ at weight $\nu-\mu$ contained in $t^{\nu}\overline{\operatorname{Gr}_{-\mu}}$.

As observed in [A], $\pi^{-1}([\nu])=\overline{\operatorname{Gr}_{\lambda}}\cap t^{\nu}\overline{\operatorname{Gr}_{-\mu}}$. Therefore $\pi^{-1}([\nu])$ carries a natural $H:=G(\mathcal{O})\cap t^{\nu}G(\mathcal{O})t^{-\nu}$ action on the left. Note that $H$ is connected for the following reason: any $x(t)\in H$ has a path $x(st)$ connecting it to $x(0)$ as $s$ varies from $1$ to [math]. This gives a retraction of $H$ onto $P_{\nu}\subset H$, and $P_{\nu}$ is path-connected (as before, $P_{\nu}$ is the parabolic subgroup of $G$ containing $B^{-}$ and $L_{\nu}$). Therefore each irreducible component of $\pi^{-1}([\nu])$ is $H$-stable.

Theorem 4.1**.**

Let $\lambda,\mu$ be dominant coweights. If $\nu=v(\lambda+w\mu)$ is dominant for some $v,w\in W$, then $V(\nu)$ appears in $V(\lambda)\otimes V(\mu)$ with multiplicity at least $1$. In fact, there is a unique MV-cycle $\overline{\operatorname{Gr}_{\lambda}}$ at weight $\nu-\mu$ contained in $t^{\nu}\overline{\operatorname{Gr}_{-\mu}}$ which contains $[v\lambda]$ (equivalently, contains $[qv\lambda]$ for all $q\in W_{\nu}$).

Proof.

The point $[v\lambda]$ is clearly contained in $\pi^{-1}([\nu])$, since $[-v\lambda+\nu]=[vw\mu]\in\operatorname{Gr}_{\mu}$.

Claim $H.[v\lambda]$ has dimension $\langle\rho,\lambda+\mu-\nu\rangle$.

Proof Exactly analogous to the previous dimension calculation.

Therefore the closure of $H.[v\lambda]$ gives an irreducible component of $\pi^{-1}([\nu])$ of the right dimension, so contributing to the multiplicity of $V(\nu)$ in $V(\lambda)\otimes V(\mu)$.

For uniqueness: if $A$ is any other irreducible component, $[v\lambda]\in A$ implies $H.[v\lambda]\subseteq A$, which forces $A=\overline{H.[v\lambda]}$.

Notably, any lift of any $q\in W_{\nu}$ to $G$ satisfies $q\in t^{\nu}G(\mathcal{O})t^{-\nu}$; therefore $[qv\lambda]\in H.[v\lambda]$. ∎

In similar style, the $m_{\lambda,\mu,w}$-many components produced as in the refinement are simply the $H$-orbits of the $[r\lambda]$s, where $\nu=r(\lambda+u\mu)$ as $u$ varies in $W_{\lambda}\backslash W/W_{\mu}$.

Corollary 4.2**.**

If $\nu=\lambda+w\mu$ is dominant, the multiplicity of $V(\nu)$ in $V(\lambda)\otimes V(\mu)$ is exactly 1.

Proof.

The cycle $A=\overline{H.[\lambda]}$ contributes $1$ to the multiplicity count. Since every MV-cycle of $\overline{\operatorname{Gr}_{\lambda}}$ at weight $\nu-\mu$ contained in $t^{\nu}\overline{\operatorname{Gr}_{-\mu}}$ must contain $[\lambda]$ and be $H$-stable, $A$ must be the only such cycle. ∎

5. The converse fails

The entire basis of this work is a very strange phenomenon: for PRV triples $\lambda,\mu,\nu$, there exist irreducible components of $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ containing a dense $G(\mathcal{O})$-orbit (equivalently, there exist MV-cycles in $\mathcal{F}_{\nu}$ containing a dense $H$-orbit). One could ask: given an irreducible top component of a cyclic convolution variety $\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ that contains a dense $G(\mathcal{O})$-orbit, is it true that $\lambda,\mu,\nu$ is a PRV triple? The answer turns out to be false:

Theorem 5.1**.**

There exist $G,\lambda,\mu,\nu$ and an irreducible component $A\subset\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$ of dimension $\langle\rho,\lambda+\mu+\nu\rangle$ such that

1. (1)

$A=\overline{G(\mathcal{O})x}$* for some $x$;* 2. (2)

there are no elements $v,w\in W$ making $\nu=v(-\lambda-w\mu)$ true.

Proof.

Here is an example: take $G=SL_{2}$, $\lambda=\mu=\nu=\alpha^{\vee}$, the single positive coroot. Criterion (2) is easy to verify: $w\mu=\pm\alpha^{\vee}$ for any $w\in W$, and $v^{-1}\nu=\pm\alpha^{\vee}$ for any $v\in W$. But

[TABLE]

is not true for any choices of $+$,$-$.

As for (1): let y=\left[\begin{array}[]{cc}1&t\\ 0&1\end{array}\right]t^{\alpha^{\vee}}.

Claim $x:=([\alpha^{\vee}],\bar{y},[0])\in\operatorname{Gr}_{G,c(\lambda,\mu,\nu)}$.

Proof We have

[TABLE]

the second line follows from

[TABLE]

and the third from

[TABLE]

Claim The dimension of $SL_{2}(\mathcal{O}).x$ is $\langle\rho,3\alpha^{\vee}\rangle=\langle\alpha/2,3\alpha^{\vee}\rangle=3$.

Proof The stabilizer of $x$ has Lie algebra

[TABLE]

we now try to express this vector space more explicitly.

Let $e,f,h$ be the standard basis of $\mathfrak{sl}_{2}(\mathbb{C})$; then

[TABLE]

and

[TABLE]

Of course, $\operatorname{Ad}_{y}\mathfrak{sl}_{2}=\operatorname{Ad}_{z}\operatorname{Ad}_{t^{\alpha^{\vee}}}\mathfrak{sl}_{2}$, where z=\left[\begin{array}[]{cc}1&t\\ 0&1\end{array}\right]. One calculates

[TABLE]

Let $X\in\mathfrak{sl}_{2}(\mathcal{O})\cap\operatorname{Ad}_{t^{\alpha^{\vee}}}\mathfrak{sl}_{2}(\mathcal{O})$ be arbitrary: $X=p_{e}e+p_{h}h+p_{f}f$, where $\operatorname{val}_{t}(p_{e})\geq 2$, $\operatorname{val}_{t}(p_{h})\geq 0$, and $\operatorname{val}_{t}(p_{f})\geq 0$ (as usual, $\operatorname{val}_{t}(0)=\infty$).

Now $X\in\operatorname{Ad}_{y}\mathfrak{sl}_{2}(\mathcal{O})$ if and only if $\operatorname{Ad}_{z^{-1}}X\in\operatorname{Ad}_{t^{\alpha^{\vee}}}\mathfrak{sl}_{2}(\mathcal{O})$. As

[TABLE]

this is if and only if $\operatorname{val}_{t}(p_{h})\geq 1$ (if $\operatorname{val}_{t}(p_{h})=0$, then the $e$-coefficient has $t$-valuation $1$ since $\operatorname{val}_{t}(p_{e}-t^{2}p_{f})\geq 2$.)

Therefore $L=t^{2}e(\mathcal{O})\oplus th(\mathcal{O})\oplus f(\mathcal{O})$, in which case

[TABLE]

and the latter has dimension $3$. So $\dim SL_{2}(\mathcal{O}).x=3$.

The usual arguments then apply: $SL_{2}(\mathcal{O}).x$ is irreducible of maximal dimension; therefore its closure is an irreducible component. ∎

Appendix: A more general framework

by Prakash Belkale and Joshua Kiers111We thank N. Fakhruddin and S. Kumar for useful discussions.

Let $H\to G$ be an embedding of complex reductive algebraic groups, and assume maximal tori and Borel subgroups are chosen such that $T_{H}\subseteq T_{G}$ and $B_{H}\subseteq B_{G}$. A priori, there is not a map $H^{\vee}\to G^{\vee}$ of Langlands dual groups; i.e., taking Langlands dual is not functorial. However, for any collection of coweights $\lambda_{1},\ldots,\lambda_{s}$ for $T_{H}$ dominant w.r.t. $B_{H}$, there is a morphism of cyclic convolution varieties

[TABLE]

where for each $i$, the “transfer” $\lambda_{i}^{\prime}:=w_{i}\lambda_{i}$ is the unique $G$-Weyl group translate of $\lambda_{i}$, viewed as a coweight of $T_{G}$, which is dominant w.r.t. $B_{G}$. The morphism is just the embedding $H(\mathcal{K})/H(\mathcal{O})\to G(\mathcal{K})/G(\mathcal{O})$ in each factor; one easily verifies it is well-defined.

Therefore it is clear that $\operatorname{Gr}_{H,c(\vec{\lambda})}\neq\emptyset\implies\operatorname{Gr}_{G,c(\vec{\lambda}^{\prime})}\neq\emptyset$.

Question 5.2**.**

Under what conditions on $H,G$ is true that

[TABLE]

for every tuple $(\lambda_{1},\ldots,\lambda_{s})$?

Equivalently, under what conditions on $H,G$ is it the case that if $\operatorname{Gr}_{H,c(\vec{\lambda})}$ has top-dimensional components then $\operatorname{Gr}_{G,c(\vec{\lambda}^{\prime})}$ does, too?

We note that consideration of mappings of “dual groups” is an important theme in the Langlands program (cf. the functoriality conjecture [Gel, Conjecture 3]).

The weaker implication

[TABLE]

does hold; this is because the Hermitian eigenvalue cones for $H^{\vee}$ and $H$ are isomorphic, as are those for $G^{\vee}$ and $G$, see [KLM]*Theorem 1.8, and there is a map between the Hermitian eigenvalue cones for $H$ and $G$ since there is a compatible mapping of maximal compact subgroups, see [BK]. Therefore implication (5.1) always holds when $G$ is of type $A$ [KT] or types $D_{4},D_{5},D_{6}$ [KKM, Ki] by saturation. Here we note that $\operatorname{Gr}_{G,c(\vec{\lambda}^{\prime})}\neq\emptyset$ implies that $\sum\lambda_{i}^{\prime}$ is in the coroot lattice for $G$ which equals the root lattice of ${G^{\vee}}$.

Setting $s=3$, the PRV theorem can be phrased as a partial answer to this question: if $H=T_{G}$ is a maximal torus of $G$, then (under no further conditions) implication (5.1) always holds. Indeed, $(V(\lambda_{1})\otimes V(\lambda_{2})\otimes V(\lambda_{3}))^{T^{\vee}}\neq(0)$ if and only if $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$; therefore the $\lambda_{i}^{\prime}$ satisfy $\lambda_{1}^{\prime}+w\lambda_{2}^{\prime}+v\lambda_{3}^{\prime}=0$ for suitable $w,v\in W$ and PRV says that $(V(\lambda_{1}^{\prime})\otimes V(\lambda_{2}^{\prime})\otimes V(\lambda_{3}^{\prime}))^{G^{\vee}}\neq(0)$.

A series of instances where the implication (5.1) holds can be found in [HS, §2]. In these examples $H$ is the subgroup of fixed points of a group $G$ under a diagram automorphism. Further, in each of these situations $H$ is of adjoint type.

When $H=PSL(2)$ and $G$ is arbitrary, implication (5.1) holds with no conditions. This follows from the linearity of the map $(\lambda_{i})\mapsto(\lambda_{i}^{\prime})$ when the $\lambda_{i}$ are each coweights of $SL(2)$ and from the special form of the Hilbert basis of the tensor cone for $SL(2)$: they are $(\omega,\omega,0)$ and permutations, so their transfers are $(\lambda^{\prime},\lambda^{\prime},0)$ for some $\lambda^{\prime}$. Since $(N\lambda^{\prime},N\lambda^{\prime},0)$ have invariants for some $N$ by (5.2), $N\lambda^{\prime}$ is self-dual; therefore $\lambda^{\prime}$ is also.

When $H=PSp(4)$ (type $C_{2}$) and $G=PSp(4m)$, we have checked that the transfer property (5.1) holds. To do this, we establish that the transfer map on dominant weights is linear. Then we identify a finite generating set for the tensor semigroup for $PSp(4)$, using a result of Kapovich and Millson [KM]. Finally we check the transfer property on this set.

However, we can exhibit the failure of (5.1) when $H=SL(2)$ and $G=SO(5)$, the map being the standard $SL(2)$ embedding corresponding to the root $\alpha_{1}$. Therefore some conditions on $H,G$ must be necessary; perhaps is suffices to assume that $Z(H^{\prime})$ maps into $Z(G)$ where $H^{\prime}=[H,H]$ is the semisimple part of $H$, and $Z(\cdot)$ denotes the center. This includes the PRV case (since $H^{\prime}=1$), as well as any case where $H$ is of adjoint type; it furthermore excludes the counterexample with $SL(2)\subseteq SO(5)$.