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


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.

This is a preview of subscription content, log in to check access.


  1. 1.

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


  1. 1.

    Anderson, J.E.: A polytope calculus for semisimple groups. Duke Math. J. 116(3), 567–588 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigenvsheaves. http://www.math.uchicago.edu/~arinkin/langlands/

  3. 3.

    Belkale, P., Kumar, S.: Eigencone, saturation, and Horn problems for symplectic and odd orthogonal groups. J. Alg. Geom. 19, 199–242 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Borel, A., Tits, J.: Groupes réductifs. Publ. Math. IHES 27, 55–150 (1965)

    MATH  Article  Google Scholar 

  5. 5.

    Gelbart, S.: An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (N.S.) 10(2), 177–219 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Ginzburg, V., Perverse sheaves on a loop group and Langlands duality, math.AG/9511007

  7. 7.

    Haines, T.J.: Equidimensionality of convolution morphisms and applications to saturation problems. Adv. in Math. 207(1), 297–327 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Haines, T.J.: Structure constants for Hecke and representation rings. Int. Math. Res. Not. IMRN 39, 2103–2119 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Hong, J., Shen, L.: Tensor invariants, saturation problems, and Dynkin automorphisms, vol. 285 (2015)

  10. 10.

    Kamnitzer, J.: Hives and the fibres of the convolution morphism. Selecta Math. N.S. 13(3), 483–496 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Kapovich, M., Kumar, S., Millson, J.: The eigencone and saturation for Spin(8). Pure Appl Math. Q. 5(2), 755–780 (2009). Special Issue: In honor of Friedrich Hirzebruch. Part 1

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Kapovich, M., Leeb, B., Millson, J.: The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra. Amer. Math Soc., Providence (2008)

  13. 13.

    Kapovich, M., Millson, J.J.: Structure of the tensor product semigroup. Asian J. Math. 10(3), 493–540 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Kiers, J.: On the saturation conjecture for Spin(2n). Exp. Math. (2019)

  15. 15.

    Knutson, A., Tao, T: The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc. 12 (4), 1055–1090 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Kostant, B.: A formula for the multiplicity of a weight. Trans. Am. Math. Soc. 93, 53–73 (1959)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Kumar, S.: Kac-Moody Groups, their Flag Varieties and Representation Theory. Birkhaäuser Boston (2002)

  18. 18.

    Kumar, S.: Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture. Invent. Math. 93(1), 117–130 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Kumar, S.: Tensor product decomposition. In: Proceedings of the International Congress of Mathematicians, 3, pgs. 1226–1261 (2010)

  20. 20.

    Kumar, S.: A refinement of the PRV conjecture. Invent. Math. 97 (2), 305–311 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Kumar, S., Leeb, B., Millson, J.: The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type. Contemporary Mathematics 332, 171–195 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Kushwaha, M.S., Raghavan, K.N., Viswanath, S.: A study of Kostant-Kumar modules via Littelmann paths. arXiv:1905.05302 (2019)

  23. 23.

    Littelmann, P.: A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math. 116(1-3), 329–346 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Lusztig, G.: Singularities, character formulas and a q-analog of weight multiplicities. Asterisqué 101–102, 208–229 (1983)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Mathieu, O.: Construction d’un groupe de Kac-Moody et applications. Compositio Math. 69(1), 37–60 (1989)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Milne, J.: Algebraic Groups: the Theory of Group Schemes of Finite Type over a Field. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  27. 27.

    Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math. 166, 95–143 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Richarz, T.: A new approach to the Geometric Satake equivalence. Documenta Math. 19, 209–246 (2014)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Roth, M.: Reduction rules for Littlewood-Richardson coefficients. Inter. Math. Res. Not. 18, 4105–4134 (2011)

    MathSciNet  MATH  Google Scholar 

Download references


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.

Author information



Corresponding author

Correspondence to Joshua Kiers.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by: Pramod Achar

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

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

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

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

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 \(SL(2)\subseteq SO(5)\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


  • Tensor product decomposition
  • Geometric satake

Mathematics Subject Classification (2010)

  • 20G05
  • 14M15
  • 22E57