[Paper Review] Relation between two geometrically defined bases in representations of $GL_n$

Alexander Braverman, Dennis Gaitsgory+1 more|ArXiv.org|Nov 11, 2004

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper establishes that the Springer basis and the Mirković-Vilonen (MV) basis for weight spaces in irreducible representations of $GL_n$ coincide. Using geometric representation theory, the authors compare two realizations of representations via the affine Grassmannian and show equivalence through intersection cohomology computations on the Zastava space, leveraging convolution and perverse sheaf techniques in the context of $GL_n$-representations over characteristic 0 fields.

ABSTRACT

Let $V$ be an irreducible representation of group $GL_n({\mathbb C})$, which appears as a submodule in $({\mathbb C}^n)^{\otimes d}$, where ${\mathbb C}^n$ 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.

Motivation & Objective

• To establish the equivalence between the Springer basis and the Mirković-Vilonen (MV) basis in the weight spaces of irreducible $GL_n$-representations.
• To relate two geometric constructions of representations: one via Springer fibers and the other via spherical perverse sheaves on the affine Grassmannian.
• To use the product of two copies of the affine Grassmannian $\operatorname{Gr}_{GL_n}$, viewed as $\operatorname{Gr}_{GL_n \times GL_n}$, to compare the two bases.
• To interpret $GL_n \times GL_n$ as a Levi subgroup of $GL_{2n}$ to facilitate the comparison via geometric Satake correspondence.
• To prove the identification of bases by computing the intersection cohomology of the Zastava space, building on results from [BFGM].

Proposed method

• Realize the irreducible representation $V^\lambda$ of $GL_n$ as a direct summand in $V^{\otimes d}$, with $\lambda$ corresponding to a dominant coweight.
• Construct the Springer basis via irreducible components of Springer fibers over nilpotent matrices with Jordan type corresponding to $\lambda$.
• Define the MV basis as the set of irreducible components of the intersection of a semi-infinite orbit $S(\mu)$ with the support of an irreducible spherical perverse sheaf $\operatorname{IC}^\lambda$ on $\operatorname{Gr}_{GL_n}$.
• Use the convolution structure on the affine Grassmannian to interpret the tensor product of representations via spherical perverse sheaves, yielding a third basis isomorphic to the Springer basis.
• Work on $\operatorname{Gr}_{GL_n \times GL_n} \simeq \operatorname{Gr}_{GL_{2n}}$ by embedding $GL_n \times GL_n$ as a Levi subgroup of $GL_{2n}$, enabling comparison of the two geometric bases.
• Apply the result from [BFGM] on the intersection cohomology of the Zastava space to conclude that the two bases coincide via isomorphism of perverse sheaves on $X^\mu \times_{X^{(d)}} \operatorname{Mod}_E^d$.

Experimental results

Research questions

• RQ1 Do the Springer basis and the Mirković-Vilonen basis coincide in the weight spaces of irreducible $GL_n$-representations?
• RQ2 Can the two geometric realizations of representations—via Springer fibers and via spherical perverse sheaves on the affine Grassmannian—be related through a common geometric framework?
• RQ3 Is the basis arising from the convolution of spherical perverse sheaves on $\operatorname{Gr}_{GL_n}$ equivalent to the Springer basis?
• RQ4 Does the intersection cohomology of the Zastava space provide a bridge between the Springer and MV bases in the $GL_n$ case?
• RQ5 Can the comparison of the two bases be reduced to a computation in the geometry of $\operatorname{Gr}_{GL_{2n}}$ via the Levi embedding $GL_n \times GL_n \subset GL_{2n}$?

Key findings

• The Springer basis and the Mirković-Vilonen basis for the weight space $V^\lambda(\mu)$ in an irreducible representation of $GL_n$ are canonically isomorphic.
• The isomorphism between the two bases is established by showing that two different constructions of the same perverse sheaf on $X^\mu \times_{X^{(d)}} \operatorname{Mod}_E^d$—one via global Springer resolution and one via convolution on the affine Grassmannian—coincide.
• The key identification arises from the fact that both constructions yield the same constant perverse sheaf on $\overset{\circ}{X}^\mu \times_{X^\mu} \operatorname{Conv}^{\overline{d}'}(\operatorname{Mod}_E)$, which is preserved under the small morphism $p_n$.
• The isomorphism of perverse sheaves $\mathfrak{q}(\mu)_!(\mathcal{F}^d_{\text{glob}})[d'(\mu)] \simeq (p_n)_!(\overline{\mathbb{Q}}_\ell[d \cdot n])$ holds both globally and fiberwise over $d \cdot x \in X^{(d)}$, confirming the basis equivalence.
• The proof relies on the smallness of the map $p_n: \operatorname{Conv}^{\overline{d}'}(\operatorname{Mod}_E) \to \operatorname{Mod}_E^d$, which ensures that the pushforward preserves the perverse t-structure and allows identification of the cohomology with the constant sheaf.
• The final identification of bases is a consequence of the intersection cohomology computation of the Zastava space from [BFGM], which provides the necessary geometric input to equate the two realizations.