A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety
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A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety Joshua Kiers1 Received: 7 November 2019 / Accepted: 8 September 2020 / © Springer Nature B.V. 2020
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. Keywords Tensor product decomposition · Geometric satake Mathematics Subject Classification (2010) 20G05 · 14M15 · 22E57
1 Introduction We give a short proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture first proven independently by Kumar in [18] and Mathieu in [25]. Our method extends to give a proof of Verma’s refined conjecture which was first proven by Kumar in [20]. We demonstrate the existence of orbits in a certain configuration variety (a geodesic triangle in the sense of [12]) to deduce, via the Geometric Satake equivalence, the representation-theoretic implications. Let G be a complex reductive group and T ⊂ B ⊂ G a fixed maximal torus and Borel ˇ denote the complex reductive Langlands dual group and Tˇ ⊂ Bˇ the corsubgroup. Let G responding dual torus and Borel subgroup. Let W be the Weyl group of G (equivalently, of ˇ The statement of the original theorem is G). ˇ with respect to B, ˇ and Theorem 1 (PRV conjecture) Let λ, μ be dominant weights for G let w ∈ W be any Weyl group element. Find v ∈ W so that ν := v(−λ − wμ) is dominant. Then ˇ (V (λ) ⊗ V (μ) ⊗ V (ν))G = 0. Presented by: Pramod Achar Joshua Kiers
[email protected] 1
University of North Carolina, Chapel Hill, NC 27599, USA
J. Kiers
Kumar proved a refinement of this theorem in [20] regarding the dimensions of the spaces of invariants. Let Wδ for any weight δ denote the stabilizer subgroup of δ in W . The stronger theorem is Theorem 2 (Refinement) Let λ, μ, ν, w be as above. Let mλ,μ,w count the number of distinct cosets u¯ ∈ Wλ \W/Wμ such that −λ−wμ and −λ−uμ are W -conjugate (equivalently, ν can be written q(−λ − uμ) for some q ∈ W ). Then ˇ
dim (V (λ) ⊗ V (μ) ⊗ V (ν))G ≥ mλ,μ,w . In particular, since mλ,μ,w ≥ 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 [8, §2], although our symmetric formulation is from [10, ˇ w.r.t. B; ˇ these induce §1]. Let λi , i = 1, . . . , s be a collection of dominant weights for G dominant coweights of G w.r.t. B. Set K = C((t)), O = C[[t]]. The groups G(O) and G(K) are, respectively, an affine group scheme and an affine group ind-scheme over C. The homogeneous space G(K)/G(O), called the affine Grassmannian, naturally has the structure of an ind-variety. Each cocharacter λ : C× → T induces an element t λ of G(K); denote b
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