A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety

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.

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Notes

  1. 1.

    We thank N. Fakhruddin and S. Kumar for useful discussions.

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Acknowledgements

I thank Shrawan Kumar, Prakash Belkale, Joel Kamnitzer, and Marc Besson for helpful discussions and suggestions. I also thank a referee for some useful comments and corrections.

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Appendix: A More General Framework

Appendix: A More General Framework

by Prakash Belkale and Joshua KiersFootnote 1

Let HG 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 HG of Langlands dual groups; i.e., taking Langlands dual is not functorial. However, for any collection of coweights λ1,…, λs for TH dominant w.r.t. BH, there is a morphism of cyclic convolution varieties

$$ {\Phi}: \text{Gr}_{H,c(\vec\lambda)} \to \text{Gr}_{G,c(\vec\lambda^{\prime})}, $$

where for each i, the “transfer” \(\lambda _{i}^{\prime }:=w_{i}\lambda _{i}\) is the unique G-Weyl group translate of λi, viewed as a coweight of TG, which is dominant w.r.t. BG. 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 \(\text {Gr}_{H,c(\vec \lambda )}\ne \emptyset \implies \text {Gr}_{G,c(\vec \lambda ^{\prime })}\ne \emptyset \).

Question 1

Under what conditions on H, G is it true that

$$ \begin{array}{@{}rcl@{}} (V(\lambda_{1})\otimes \cdots\otimes V(\lambda_{s}))^{H^{\vee}}\ne 0 \implies (V(\lambda_{1}^{\prime})\otimes \cdots\otimes V(\lambda_{s}^{\prime}))^{G^{\vee}}\ne 0 \end{array} $$
(1)

for every tuple (λ1,…, λs)?

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

We note that consideration of mappings of “dual groups” is an important theme in the Langlands program (cf. the functoriality conjecture [5, Conjecture 3]).

The weaker implication

$$ \begin{array}{@{}rcl@{}} \exists N \text{ s.t. }(V(N\lambda_{1})\otimes &&\cdots\otimes V(N\lambda_{s}))^{H^{\vee}}\ne 0 \\ &&\implies \exists N^{\prime} \text{ s.t. } (V(N^{\prime}\lambda_{1}^{\prime})\otimes \cdots\otimes V(N^{\prime}\lambda_{s}^{\prime}))^{G^{\vee}}\ne 0 \end{array} $$
(2)

does hold; this is because the Hermitian eigenvalue cones for H and H are isomorphic, as are those for G and G, see [21, 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 [3]. Therefore implication (1) always holds when G is of type A [15] or types D4, D5, D6 [11, 14] by saturation. Here we note that \(\text {Gr}_{G,c(\vec \lambda ^{\prime })}\ne \emptyset \) implies that \(\sum \lambda _{i}^{\prime }\) is in the coroot lattice for G which equals the root lattice of G.

Setting s = 3, the PRV theorem can be phrased as a partial answer to this question: if H = TG is a maximal torus of G, then (under no further conditions) implication (1) always holds. Indeed, \((V(\lambda _{1})\otimes V(\lambda _{2})\otimes V(\lambda _{3}))^{T^{\vee }}\ne 0\) if and only if λ1 + λ2 + λ3 = 0; therefore the \(\lambda _{i}^{\prime }\) satisfy \(\lambda _{1}^{\prime }+w\lambda _{2}^{\prime }+v\lambda _{3}^{\prime } = 0\) for suitable w, vW and PRV says that \((V(\lambda _{1}^{\prime })\otimes V(\lambda _{2}^{\prime })\otimes V(\lambda _{3}^{\prime }))^{G^{\vee }}\ne 0\).

A series of instances where the implication (1) holds can be found in [9, §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 (1) holds with no conditions. This follows from the linearity of the map \((\lambda _{i})\mapsto (\lambda _{i}^{\prime })\) when the λi are each coweights of SL(2) and from the special form of the Hilbert basis elements of the tensor cone for SL(2): they are (ω, ω, 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 (2), \(N\lambda ^{\prime }\) is self-dual; therefore \(\lambda ^{\prime }\) is also.

When H = PSp(4) (type C2) and G = PSp(4m), we have checked that the transfer property (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 [13]. Finally we check the transfer property on this set.

However, we can exhibit the failure of (1) when H = SL(2) and G = SO(5), the map being the standard SL(2) embedding corresponding to the root α1. Therefore some conditions on H, G must be necessary; perhaps it suffices to assume that \(Z(H^{\prime })\) maps into Z(G) where \(H^{\prime }=[H,H]\) is the semisimple part of H, and Z(⋅) 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 \(SL(2)\subseteq SO(5)\).

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Kiers, J. A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety. Algebr Represent Theor (2020). https://doi.org/10.1007/s10468-020-09992-8

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Keywords

  • Tensor product decomposition
  • Geometric satake

Mathematics Subject Classification (2010)

  • 20G05
  • 14M15
  • 22E57