LS galleries, the path model, and MV cycles

By S. Gaussent, P. Littelmann. 2003

123 citations recorded by the source corpus.

We give an interpretation of the path model of a representation [18] of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology of ˆ Σ(λ) has a basis given by Bia lynicki–Birula cell’s, which are indexed by the T –fixed points in ˆ Σ(λ). Now the points of ˆ Σ(λ) can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T –fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with G(C[[t]]).λ (identified with an open subset of ˆ Σ(λ)), and we show that the closures of the strata associated to LS-galleries are exactly the MV–cycles [24], which form a basis of the representation V (λ) for the Langland’s dual group G ∨ .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aeb9853984123543fc724586efd4cd3c14ce98e2

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Buildings, spiders, and geometric Satake (2011, background): “For a detailed description of affine buildings from this perspective, see [10].”
Mirkovic-Vilonen cycles and polytopes (2005, methodology): “ics of representations of G∨. Because of this connection to the representation theory, some attempts have been made to give an explicit description of the MV cycles and polytopes. Gaussent-Littelmann [GL] associated an MV cycle to each Littelmann path. However, their method did not give equations cu”
Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra (2005, methodology): “his was shown by Rapoport for the case of spherical Hecke algebras of a reductive group over a local ﬁeld [Rap00]. Remark 4.9. For regular λ the deﬁnition of galleries coincides with the one given in [GL05]. Instead of using generalized alcoves they regard galleries of alcoves together with an initi”
ON THE COMBINATORICS OF CRYSTAL GRAPHS, I. LUSZTIG'S INVOLUTION (2005, unknown): “For instance, a straight line initial path leads to the Lakshmibai-Seshadri paths (LS paths) [16]; these were introduced before Littelmann’s work, in the context of standard monomial theory [16].”