Relation between two geometrically defined bases in representations of GL_n

2004/11/11 by Alexander Braverman, Braverman, Alexander, Dennis Gaitsgory +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.math/0411252

openalex publication_date 2004/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V be an irreducible representation of group GL_n(\mathbb C), which appears as a submodule in (\mathbb Cⁿ)^{⊗ d}, where \mathbb Cⁿ is the tautological n-dimensional representation of GL_n, and d is a non-negative integer. On the one hand, following refs [Gi] and [BG] one can produce a basis in V using irreducible components of Sringer fibers over a nilpotent matrix in \mathfrak gl_d, whose Jordan blocks correspond to the highest weight of V. On the other hand, one can produce a basis in V by Mirković-Vilonen cycles, a construction that works for an arbitrary reductive group G. In this note we prove that the resulting to bases coincide.

Citations

• Geometric Langlands duality and representations of algebraic groups over commutative rings

Related

• Bases of tensor products and geometric Satake correspondence
• Lectures on geometric constructions of the irreducible representations of GL_n
• Mirković-Vilonen basis in type A₁
• Représentations modulo p de GL(m,D), D algèbre à division sur un corps local
• Symmetrization of representations of GL_N