Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

Morier-Genoud, Sophie

doi:10.1090/s0002-9947-07-04216-x

Cited by 12 publications

(25 citation statements)

References 16 publications

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“…This lemma shows that, when W is a Weyl group, then (y 1 , · · · , y q ) are the Lusztig coordinates with respect to the decomposition i * of the image of the path η with string coordinates (x 1 , · · · , x q ) with respect to the decomposition i under the Schutzenberger involution, where i * is obtained from i by the mapα = −w 0 α (see Morier-Genoud [29], Cor. 2.17).…”

Section: Proof Of Theorem 52mentioning

confidence: 99%

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Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Biane

Bougerol

O’Connell

2009

Advances in Mathematics

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We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations.By theorem 5.5, J λ (z) = E(e z,η(s) |σ(P w0 η(a), a ≤ s) when P w0 η(s) = λ and, since τ s η and η have the same law, J µ (z) = E(e z,τsη(t) |σ(P w0 τ s η(b), b ≤ t)) when P w0 τ s η(t) = µ. Therefore E(e z,η(s+t) |G s,t ) = J λ (z)J µ (z).On the other hand, by lemma 4.12, G s,t is contained in σ(P w0 η(r), r ≤ s + t), thus E(e z,η(s+t) |G s,t ) = E(E(e z,η(s+t) |σ(P w0 η(r), r ≤ s + t))|G s,t ) = E(J z (P w0 η(s + t))|G s,t ). It thus follows from theorem 5.15 thatTherefore, for all z ∈ V * ,By injectivity of the Fourier-Laplace transform this implies thatThe positive product formula gives a positive answer to a question of Rösler [35] for the radial Dunkl kernel. It shows that one can generalize the structure

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Section: Proof Of Theorem 52mentioning

confidence: 99%

“…We will establish an analogous property for the analogue of the Schützenberger involution defined in [2] for finite Coxeter groups. The crystallographic case has been recently investigated by Henriques and Kamnitzer [15], [16], and Morier-Genoud [29].…”

Section: Schützenberger Involutionmentioning

confidence: 99%

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Biane

Bougerol

O’Connell

2009

Advances in Mathematics

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“…Morier-Genoud [40] has applied the ideas of [5] to study generalizations of the Schützenberger involution for highest weight modules of semisimple groups. Though not addressing the above question, many of the quantities obtained are similar to those of Section 5.…”

Section: Generic Evaluation Of the Exponential Sum At Prime Powersmentioning

confidence: 99%

Whittaker Coefficients of Metaplectic Eisenstein Series

Brubaker

Friedberg

2015

Geom. Funct. Anal.

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Abstract. We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However they are not Eulerian and the standard methods to compute them in the reductive case do not apply to covers. For "cominuscule" maximal parabolics, we give an explicit description of the coefficients as Dirichlet series whose arithmetic content is expressed in an exponential sum. The exponential sum is then shown to satisfy a twisted multiplicativity, reducing its determination to prime power contributions. These, in turn, are connected to Lusztig data for canonical bases on the dual group using a result of Kamnitzer. The exponential sum at prime powers is shown to simplify for generic Lusztig data. At the remaining degenerate cases, the exponential sum seems best expressed in terms of Gauss sums depending on string data for canonical bases, as shown in a detailed example in GL 4 . Thus we demonstrate that the arithmetic part of metaplectic Whittaker coefficients is intimately connected to the relations between these two expressions for canonical bases.

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“…It is also worth noting that there are well-known bijections between the Lusztig parametrization, string parametrization, and semistandard Young tableaux in type A r . More details may be found in [27,29].…”

Section: Introductionmentioning

confidence: 99%

Young Tableaux, Canonical Bases, and the Gindikin-Karpelevich Formula

Lee¹,

Salisbury²

2014

Journal of the Korean Mathematical Society

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Abstract. A combinatorial description of the crystal B(∞) for finitedimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux. IntroductionThe Gindikin-Karpelevich formula is a p-adic integration formula proved by Langlands in [18]. He named it the Gindikin-Karpelevich formula after a similar formula originally stated by Gindikin and Karpelevich [5] in the case of real reductive groups. The formula also appears in Macdonald's work [25] on p-adic groups and affine Hecke algebras.Let G be a split semisimple algebraic group over a p-adic field F with ring of integers o F , and suppose the residue field o F /πo F of F has size t, where π is a generator of the unique maximal ideal in o F . Choose a maximal torus T of G contained in a Borel subgroup B with unipotent radical N , and let N − be the opposite group to N . We have B = T N . The group G(F ) has a decomposition G(F ) = B(F )K, whereis the modular character of B and τ is extended to B(F ) to be trivial on N (F ). The function f• is called the standard spherical vector corresponding to τ .Let G ∨ be the Langlands dual of G with the dual torus T ∨ . The set of coroots of G is identified with the set of roots of G ∨ and will be denoted by Φ. Finally, let z be the element of the dual torus T ∨ , corresponding to τ via the Satake isomorphism.Theorem 0.1 (Gindikin-Karpelevich formula, [18]). Given the setting above, we havewhere Φ + is the set of positive roots of G ∨ .Let g be the Lie algebra of G ∨ , and let B(∞) be the crystal basis of the negative part U − q (g) of the quantum group U q (g). Then Φ is the root system of g as well. In recent work, the integral in the Gindikin-Karpelevich formula has been evaluated using Kashiwara's crystal basis or Lusztig's canonical basis. D. Bump and M. Nakasuji where nz(φ i (b)) is the number of nonzero entries in the Lusztig parametrization φ i (b) of b with respect to a reduced expression i of the longest Weyl group element. The purpose of this paper is to describe the sum in (0.2) in a combinatorial way using Young tableaux. Since the canonical basis B is the same as Kashiwara's global crystal basis, we may replace B with B(∞). The associated crystal structure on B may be described in terms of the Lusztig parametrization [2,24] or the string parametrization [1,13,22], and there are formulas relating the two given by Berenstein and Zelevinsky in [2]. Much work has been done on realizations of crystals (e.g., [8,9,15,16,21]). In the case of B(∞) for finitedimensional simple Lie algebras, J. Hong and H. Lee used marginally large semistandard Young tableaux to obtain a realization of crystals [7]. We will use their marginally large semistandard Young tableaux realization of B(∞) to write the right-hand side of (0.2) as a sum over a set T (∞) ...

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Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

Cited by 12 publications

References 16 publications

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Continuous crystal and Duistermaat–Heckman measure for Coxeter groups

Whittaker Coefficients of Metaplectic Eisenstein Series

Young Tableaux, Canonical Bases, and the Gindikin-Karpelevich Formula

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