## 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.
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 Kiers^{Footnote 1}

Let *H* → *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*^{∨}→ *G*^{∨} of Langlands dual groups; i.e., taking Langlands dual is not functorial. However, for any collection of coweights *λ*_{1},…, *λ*_{s} for *T*_{H} dominant w.r.t. *B*_{H}, there is a morphism of cyclic convolution varieties

where for each *i*, the “transfer” \(\lambda _{i}^{\prime }:=w_{i}\lambda _{i}\) is the unique *G*-Weyl group translate of *λ*_{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 \(\text {Gr}_{H,c(\vec \lambda )}\ne \emptyset \implies \text {Gr}_{G,c(\vec \lambda ^{\prime })}\ne \emptyset \).

###
**Question 1**

Under what conditions on *H*, *G* is it true that

for every tuple (*λ*_{1},…, *λ*_{s})?

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

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 *D*_{4}, *D*_{5}, *D*_{6} [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* = *T*_{G} 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 *λ*_{1} + *λ*_{2} + *λ*_{3} = 0; therefore the \(\lambda _{i}^{\prime }\) satisfy \(\lambda _{1}^{\prime }+w\lambda _{2}^{\prime }+v\lambda _{3}^{\prime } = 0\) for suitable *w*, *v* ∈ *W* 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* = *P**S**L*(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 *λ*_{i} are each coweights of *S**L*(2) and from the special form of the Hilbert basis elements of the tensor cone for *S**L*(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* = *P**S**p*(4) (type *C*_{2}) and *G* = *P**S**p*(4*m*), 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 *P**S**p*(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* = *S**L*(2) and *G* = *S**O*(5), the map being the standard *S**L*(2) embedding corresponding to the root *α*_{1}. 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