Mirabolic Hecke Algebra: Structure & Dualities

• Mirabolic Hecke algebra is defined as a convolution algebra on flag pairs and a vector, generalizing the ordinary type-A Hecke algebra.
• Its algebraic presentation introduces additional generators and mixed relations, including a key idempotent that encodes the mirabolic datum.
• The structure underlies dualities such as mirabolic Schur–Weyl and Howe duality, leading to refined insights in representation theory.

The mirabolic Hecke algebra is the type $A$ Hecke-theoretic object obtained by adjoining a vector to the usual double-flag geometry. In the finite-field model, with $G=GL_d(\mathbb F_q)$, complete flag variety $\mathcal Y$, and vector space $V=\mathbb F_q^d$, it is the convolution algebra of $G$-invariant functions on $\mathcal Y\times \mathcal Y\times V$. Algebraically it may be presented by generators $T_0,T_1,\dots,T_{d-1}$, where $T_1,\dots,T_{d-1}$ satisfy the ordinary type-$A$ Hecke relations and $T_0$ encodes the additional mirabolic datum. In the recent literature the same algebra appears under the notations $G=GL_d(\mathbb F_q)$0, $G=GL_d(\mathbb F_q)$1, and $G=GL_d(\mathbb F_q)$2, depending on normalization and base ring (Fan et al., 2024, Rosso, 2013, Wan, 28 Feb 2026).

Notation	Setting	Source
$G=GL_d(\mathbb F_q)$3	generic mirabolic Hecke algebra via convolution on $G=GL_d(\mathbb F_q)$4	(Fan et al., 2024)
$G=GL_d(\mathbb F_q)$5	Rosso’s generic mirabolic Hecke algebra	(Rosso, 2013)
$G=GL_d(\mathbb F_q)$6	arbitrary-commutative-ring presentation	(Wan, 28 Feb 2026)

1. Geometric origin and orbit combinatorics

The adjective “mirabolic” refers to the mirabolic subgroup, namely the stabilizer of a nonzero vector. A basic geometric principle recalled in the literature is that, for a $G=GL_d(\mathbb F_q)$7-variety $G=GL_d(\mathbb F_q)$8, $G=GL_d(\mathbb F_q)$9-orbits on $\mathcal Y$0 correspond bijectively to $\mathcal Y$1-orbits on $\mathcal Y$2. In the finite-field incarnation used for mirabolic Hecke algebras, this principle becomes a convolution theory on spaces of the form

$\mathcal Y$3

where $\mathcal Y$4 is a partial flag variety and $\mathcal Y$5 is the complete flag variety (Fan et al., 2024).

For the Hecke side one uses complete flags. The $\mathcal Y$6-orbits on $\mathcal Y$7 are finite. They are parametrized by pairs $\mathcal Y$8, where $\mathcal Y$9 and $V=\mathbb F_q^d$0 satisfies a monotonicity condition; equivalently, $V=\mathbb F_q^d$1 may be written as a strictly decreasing sequence

$V=\mathbb F_q^d$2

such that

$V=\mathbb F_q^d$3

The characteristic function of the orbit corresponding to $V=\mathbb F_q^d$4 is denoted $V=\mathbb F_q^d$5. The elements $V=\mathbb F_q^d$6 attached to simple transpositions and $V=\mathbb F_q^d$7 supply the basic generators of the algebra (Fan et al., 2024).

The convolution product is defined by summing over an intermediate complete flag and an intermediate vector: $V=\mathbb F_q^d$8 This product is associative, and the $V=\mathbb F_q^d$9-invariant characteristic functions $G$0 form a basis. In this sense the mirabolic Hecke algebra is the direct analogue of the Iwahori–Hecke algebra of type $G$1, except that the geometry has been enlarged from pairs of flags to triples consisting of two flags and a vector (Rosso, 2013).

The extra vector is not a superficial modification. In the non-mirabolic setting orbit data are controlled by relative position of flags alone; in the mirabolic setting the vector produces a second layer of incidence data. On partial-flag varieties this is encoded by decorated matrices, and for $G$2 their behavior under convolution is explicitly described as more intricate than in the rank-one case (Fan et al., 2024).

2. Algebraic presentations

In Solomon’s presentation, the mirabolic Hecke algebra is the associative algebra generated by

$G$3

with relations

$G$4

the ordinary type-$G$5 braid relations for $G$6,

$G$7

and the two mixed relations

$G$8

$G$9

In the normalization $\mathcal Y\times \mathcal Y\times V$0, these are the relations quoted for $\mathcal Y\times \mathcal Y\times V$1 over $\mathcal Y\times \mathcal Y\times V$2 (Rosso, 2013, Fan et al., 2024).

A useful reformulation introduces the idempotent

$\mathcal Y\times \mathcal Y\times V$3

Then the algebra is generated by $\mathcal Y\times \mathcal Y\times V$4 with the ordinary Hecke relations for $\mathcal Y\times \mathcal Y\times V$5, together with

$\mathcal Y\times \mathcal Y\times V$6

and

$\mathcal Y\times \mathcal Y\times V$7

$\mathcal Y\times \mathcal Y\times V$8

This presentation makes the “one extra idempotent” structure completely explicit and is the form most directly comparable with cyclotomic Hecke algebras (Rosso, 2013).

Wan gives a further presentation over an arbitrary commutative ring $\mathcal Y\times \mathcal Y\times V$9 with invertible $T_0,T_1,\dots,T_{d-1}$0, replacing $T_0,T_1,\dots,T_{d-1}$1 by a family of idempotents $T_0,T_1,\dots,T_{d-1}$2. The first is

$T_0,T_1,\dots,T_{d-1}$3

and recursively

$T_0,T_1,\dots,T_{d-1}$4

These satisfy

$T_0,T_1,\dots,T_{d-1}$5

$T_0,T_1,\dots,T_{d-1}$6

With these generators Wan constructs a basis indexed by triples $T_0,T_1,\dots,T_{d-1}$7, where $T_0,T_1,\dots,T_{d-1}$8 have equal cardinality and $T_0,T_1,\dots,T_{d-1}$9 belongs to a Young subgroup $T_1,\dots,T_{d-1}$0. The basis elements are

$T_1,\dots,T_{d-1}$1

and this yields an arbitrary-base-ring presentation of $T_1,\dots,T_{d-1}$2 (Wan, 28 Feb 2026).

3. Cyclotomic realization, center, cocenter, and character theory

A fundamental structural result is that the mirabolic Hecke algebra is a quotient of the Ariki–Koike cyclotomic Hecke algebra $T_1,\dots,T_{d-1}$3. In Rosso’s formulation,

$T_1,\dots,T_{d-1}$4

where $T_1,\dots,T_{d-1}$5 is the two-sided ideal generated by the primitive idempotent corresponding to the bipartition $T_1,\dots,T_{d-1}$6. This quotient description explains why simple modules are indexed not by arbitrary bipartitions, but by those whose second component is a single column (Rosso, 2013).

More precisely, the simple modules of the generic mirabolic Hecke algebra are indexed by pairs $T_1,\dots,T_{d-1}$7, where $T_1,\dots,T_{d-1}$8 and $T_1,\dots,T_{d-1}$9; equivalently, by bipartitions of the form

$A$0

This is the representation-theoretic shadow of the fact that the mirabolic datum consists of a single vector rather than a higher-dimensional auxiliary space (Rosso, 2013).

The algebra admits mirabolic Jucys–Murphy elements. In the $A$1-presentation they are

$A$2

and they act diagonally in the seminormal basis attached to standard bitableaux. Rosso proves that the center is exactly the ring of symmetric polynomials in these elements: $A$3 This is the direct mirabolic analogue of the classical description of the center of the type-$A$4 Hecke algebra (Rosso, 2013).

Wan extends the structural picture from the center to the cocenter. He defines special elements

$A$5

indexed by partitions $A$6 of size at most $A$7, and shows that their images form a basis of the cocenter $A$8. Every basis element $A$9 admits an expansion modulo commutators in terms of these $T_0$0, with uniquely determined class polynomials $T_0$1. On this basis Wan defines the character table of $T_0$2, proves a Frobenius character formula in the ring of symmetric functions, and derives a recursive Murnaghan–Nakayama rule for the irreducible characters (Wan, 28 Feb 2026).

The Frobenius formula takes the form

$T_0$3

with Schur functions on the right and Hall–Littlewood-type symmetric functions on the left. The associated Murnaghan–Nakayama rule removes strips from $T_0$4 with explicit $T_0$5-dependent weights $T_0$6, giving a recursive computation of the mirabolic character table (Wan, 28 Feb 2026).

4. Dualities: Schur–Weyl, double centralizers, and Howe theory

The geometric role of the mirabolic Hecke algebra is most transparent in mirabolic Schur–Weyl duality. Let

$T_0$7

and

$T_0$8

Then $T_0$9 is naturally an $G=GL_d(\mathbb F_q)$00-bimodule by convolution. For $G=GL_d(\mathbb F_q)$01, the two actions form a double centralizer pair: $G=GL_d(\mathbb F_q)$02 This is the geometric mirabolic Schur–Weyl duality of type $G=GL_d(\mathbb F_q)$03 (Fan et al., 2024).

The same literature relates $G=GL_d(\mathbb F_q)$04 to the mirabolic quantum group $G=GL_d(\mathbb F_q)$05. The left $G=GL_d(\mathbb F_q)$06-action and right $G=GL_d(\mathbb F_q)$07-action on $G=GL_d(\mathbb F_q)$08 commute, and for $G=GL_d(\mathbb F_q)$09 one has

$G=GL_d(\mathbb F_q)$10

Thus the mirabolic Hecke algebra is the full commutant of the mirabolic quantum-group action on the geometric tensor space (Fan et al., 2024).

A complementary description uses the semidirect product

$G=GL_d(\mathbb F_q)$11

In this model,

$G=GL_d(\mathbb F_q)$12

where $G=GL_d(\mathbb F_q)$13 is the Borel idempotent. This realizes the mirabolic Hecke algebra as a corner algebra of the group algebra of $G=GL_d(\mathbb F_q)$14, directly paralleling the familiar type-$G=GL_d(\mathbb F_q)$15 description $G=GL_d(\mathbb F_q)$16 for the ordinary Hecke algebra. The same framework supports mirabolic Howe duality: for $G=GL_d(\mathbb F_q)$17, the bimodule

$G=GL_d(\mathbb F_q)$18

carries commuting actions of $G=GL_d(\mathbb F_q)$19 and $G=GL_d(\mathbb F_q)$20, and these actions are mutual centralizers (Fan et al., 2024).

Wan establishes a distinct, algebraic Schur–Weyl duality with the ordinary quantum group $G=GL_d(\mathbb F_q)$21. On $G=GL_d(\mathbb F_q)$22, the actions of $G=GL_d(\mathbb F_q)$23 and $G=GL_d(\mathbb F_q)$24 centralize each other, and one obtains

$G=GL_d(\mathbb F_q)$25

This places the mirabolic Hecke algebra in a see-saw with Jimbo’s classical duality for $G=GL_d(\mathbb F_q)$26 and $G=GL_d(\mathbb F_q)$27 (Wan, 28 Feb 2026).

5. Representation theory

At generic parameter the mirabolic Hecke algebra is semisimple. Rosso’s cyclotomic realization recovers Siegel’s classification of irreducibles, with simple modules indexed by pairs $G=GL_d(\mathbb F_q)$28 or, equivalently, by bipartitions $G=GL_d(\mathbb F_q)$29. Restriction and induction admit explicit branching rules, and the mirabolic Jucys–Murphy elements refine these functors in a way analogous to the ordinary Hecke-theoretic picture (Rosso, 2013).

This refinement leads to a $G=GL_d(\mathbb F_q)$30-structure on the Grothendieck group of the tower of mirabolic Hecke algebras. Rosso defines exact functors $G=GL_d(\mathbb F_q)$31 and $G=GL_d(\mathbb F_q)$32 via generalized eigenspaces of the last Jucys–Murphy element, and from these constructs operators $G=GL_d(\mathbb F_q)$33 on

$G=GL_d(\mathbb F_q)$34

The resulting module decomposes into countably many copies of the charge-$G=GL_d(\mathbb F_q)$35 level-$G=GL_d(\mathbb F_q)$36 Fock space (Rosso, 2013).

The relation to mirabolic quantum groups sharpens the representation theory further. In the $G=GL_d(\mathbb F_q)$37 case, Rosso’s mirabolic quantum $G=GL_d(\mathbb F_q)$38 yields a Schur–Weyl dictionary between simple modules of $G=GL_d(\mathbb F_q)$39 and simple modules of the mirabolic Hecke algebra, and this recovers the classification of irreducible $G=GL_d(\mathbb F_q)$40-modules. Later work for general $G=GL_d(\mathbb F_q)$41 makes the correspondence explicit: the irreducible $G=GL_d(\mathbb F_q)$42-modules $G=GL_d(\mathbb F_q)$43 are paired with irreducible mirabolic Hecke modules $G=GL_d(\mathbb F_q)$44 through mirabolic quantum Schur–Weyl duality (Rosso, 2015, Goyal et al., 8 Oct 2025).

A recurrent feature is that only bipartitions with second component a column occur. This should not be viewed as a notational accident. It reflects the geometry of a single distinguished vector and is precisely the constraint predicted by both the cyclotomic quotient $G=GL_d(\mathbb F_q)$45 and the mirabolic Schur–Weyl decomposition (Rosso, 2013).

6. Broader context, scope, and open directions

The mirabolic Hecke algebra belongs to a larger family of “mirabolic” constructions in geometric representation theory. Travkin studied convolution and Kazhdan–Lusztig-type bases on $G=GL_d(\mathbb F_q)$46; Finkelberg–Ginzburg introduced categories of mirabolic $G=GL_d(\mathbb F_q)$47-modules and related them to spherical trigonometric Cherednik algebras; and Bellamy–Ginzburg showed that the spherical trigonometric Cherednik algebra $G=GL_d(\mathbb F_q)$48 is morally a mirabolic version of the affine Hecke algebra. This suggests a conceptual continuum between finite mirabolic Hecke algebras, mirabolic character-sheaf phenomena, and Cherednik-theoretic Hamiltonian reduction, but the papers distinguish these objects carefully rather than identifying them outright (Bellamy et al., 2012).

One common misconception is that the mirabolic Hecke algebra is merely the ordinary type-$G=GL_d(\mathbb F_q)$49 Hecke algebra with an ad hoc extra idempotent. The literature does not support that simplification. The defining mixed relations, the orbit parametrization by $G=GL_d(\mathbb F_q)$50, the appearance of decorated matrices on partial-flag spaces, and the double-centralizer role of $G=GL_d(\mathbb F_q)$51 all arise from a genuine enlargement of the underlying geometry from pairs of flags to triples of two flags and a vector (Fan et al., 2024).

Another active theme is cellularity. Rosso conjectured that $G=GL_d(\mathbb F_q)$52 is a cellular algebra, proposed a strategy via the cellular basis of the cyclotomic Hecke algebra $G=GL_d(\mathbb F_q)$53, and verified the conjecture for $G=GL_d(\mathbb F_q)$54. A full proof for all $G=GL_d(\mathbb F_q)$55 was not available in that work (Rosso, 2013). Wan’s arbitrary-ring presentation and cocenter basis suggest that structural questions of this sort remain central to the subject (Wan, 28 Feb 2026).

The recent direction of the field is therefore twofold. One direction strengthens the internal structure of $G=GL_d(\mathbb F_q)$56: new presentations, cocenters, character formulas, and recursive character theory. The other embeds $G=GL_d(\mathbb F_q)$57 into larger duality frameworks: geometric mirabolic Schur–Weyl duality, mirabolic Howe duality, and algebraic duality with $G=GL_d(\mathbb F_q)$58. Together these developments establish the mirabolic Hecke algebra as a stable object of type-$G=GL_d(\mathbb F_q)$59 representation theory, with its own orbit combinatorics, central and cocentral structure, and a network of dualities paralleling—but not duplicating—the classical theory (Fan et al., 2024, Wan, 28 Feb 2026).