# An equivariant basis for the cohomology of Springer fibers

## Abstract

Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.

## 1. Introduction

This paper analyzes the equivariant cohomology of Springer fibers in Lie type A. Springer fibers are fibers of a desingularization of the nilpotent cone in Springer showed that the symmetric group acts on the cohomology of each Springer fiber, the top-dimensional cohomology is an irreducible representation, and each irreducible symmetric group representation can be obtained in this way .Reference 35Reference 36. As a consequence, Springer fibers frequently arise in geometric representation theory and algebraic combinatorics; see Reference 14Reference 15Reference 18Reference 20Reference 32Reference 34 for just a few examples.

There is also an algebraic approach to the Springer representation for as we now explain. Motivated by a conjecture of Kraft ,Reference 25, De Concini and Procesi Reference 8 gave a presentation for the cohomology of a type A Springer fiber as the quotient of a polynomial ring. Furthermore, this identification is so Springer’s representation can also be constructed as the symmetric group action on the quotient of a polynomial ring. These results were generalized to the setting of other algebraic groups by Carrell in -equivariantReference 6.

The generators of the ideal defining the presentation of the cohomology of a type A Springer fiber were further simplified by Tanisaki Reference 37. Finally, Garsia and Procesi used the aforementioned results to study the graded character of the Springer representation in Reference 19. Their work gives a linear algebraic proof that this character is closely connected to the so-called polynomials. As part of their analysis, Garsia and Procesi study a monomial basis for the cohomology ring, originally defined by De Concini and Procesi in -KostkaReference 8, with many amenable combinatorial and inductive properties. We refer to the collection of these monomials as the *Springer monomial basis*.

Let denote the algebraic group of invertible matrices with Lie algebra of matrices. Denote by the Borel subgroup of upper triangular matrices, and by its Lie algebra. Given a nilpotent matrix let , be the partition of determined by the sizes of the Jordan blocks of The flag variety of . is the quotient and the Springer fiber corresponding to is defined as the subvariety

Let denote the maximal torus of diagonal matrices in and be the Levi subgroup of block diagonal matrices with block sizes determined by the partition We may assume without loss of generality that . is in Jordan canonical form, and hence is regular in the Lie algebra of Moreover, the subtorus . acts on the Springer fiber We consider the equivariant cohomology . The goal of this manuscript is provide a combinatorial framework to study this equivariant cohomology. .

There is a known presentation for given by Kumar and Procesi Reference 26, and the equivariant Tanisaki ideal has been determined by Abe and Horiguchi Reference 1. Our work below initiates a study of which parallels the analysis of the ordinary cohomology by Garsia and Procesi in Reference 19. We define a collection of polynomials in using the combinatorics of row-strict tableaux. Since these polynomials map onto the Springer monomial basis under the natural projection map from equivariant to ordinary cohomology we call them ,*equivariant Springer monomials*. We prove that a basis of equivariant Springer monomials exists for any Springer fiber, and provide a determinant formula (see Theorem 4.5 below) for the structure constants of any element of with respect to this basis.

As an application, we use the algebraic and combinatorial framework developed in this manuscript to study the images of Schubert classes in Let . denote the inclusion of varieties, and the induced map on ordinary cohomology. We prove that for every partition there is a natural collection of Schubert classes whose images under , form an additive basis of This result appears as Theorem .5.9 in Section 5 below and Corollary 5.14 contains the equivariant version of the statement. Phrased in terms of the work of Harada and Tymoczko in Reference 22, the equivariant version of Theorem 5.9 says that there exists a successful game of *Betti poset pinball* for each type A Springer fiber. As a result, we can do computations in the (equivariant) cohomology ring more easily, as combinatorial properties of (double) Schubert polynomials are well-studied (c.f., for example, Reference 29). Bases of this kind have been used to do Schubert calculus style computations in the equivariant cohomology rings of other subvarieties of the flag variety Reference 10Reference 21; the authors will explore analogous computations for Springer fibers in future work.

Our Theorem 5.9 generalizes results of Harada–Tymoczko Reference 22 and Dewitt–Harada Reference 9 which address the case of and respectively. The main difficulty in generalizing the methods used in those papers is that the equivariant cohomology classes in , constructed via poset pinball may not satisfy upper triangular vanishing conditions (with respect to some partial ordering on the set of points of -fixed The methods used to prove Theorem ).5.9 side-step this difficulty by making use of the equivariant Springer monomials. Combining our determinantal formula for the structure coefficients of this basis with known combinatorial properties of the Schubert polynomials yields the desired result.

Let denote the Schubert variety corresponding to a permutation

In this paper, we study the polynomials

for generic

The remainder of the paper is structured as follows. The next section covers the necessary background information and notation needed in later sections, including a presentation of the equivariant cohomology of the Springer fiber due to Kumar and Procesi. The third and fourth sections of this paper establish the combinatorial groundwork for our study of

## 2. Background

As in the introduction, let

The composition

Let **Springer fiber** of

If two compositions have the same underlying partition shape, then the corresponding Springer fibers are isomorphic. However, taking different compositions corresponding to the same partition shape yields actions of different sub-tori of

Let

### 2.1. A presentation of

Recall from the introduction that

Let

where

It is well known that the

Since

is defined by

given by

Let

Note that we may also view

is naturally an

where

Since the map

### 2.2. Maps of polynomial rings

Recall that

where

This embedding induces a map

where

As in Equation 2.1, there are isomorphisms

where

and we make this identification below whenever it is convenient (and similarly for

The ring

denote the positive degree and degree zero components of

given by evaluation at

where, by Theorem 2.1,