Lusztig's Map in Representation Theory

• Lusztig’s map is a family of canonical constructions that transfer data between combinatorial or spectral parameter spaces and geometric objects, as seen in Jordan decompositions and Springer correspondences.
• It employs functorial methods to ensure compatibility with parabolic induction, theta correspondence, and categorical equivalences across finite groups, affine Hecke algebras, and reductive groups.
• The map’s applications span from classifying unipotent characters and Weyl-group representations to realizing braid symmetries, canonical bases in quantum groups, and the geometry of quiver varieties.

Searching arXiv for the cited papers and closely related work on “Lusztig’s map” to ground the article in the current literature. In modern representation theory, the expression “Lusztig’s map” is not attached to a single universally fixed construction. It denotes a family of maps, correspondences, and functorial identifications introduced by Lusztig, or later canonical refinements of Lusztig’s constructions, that transfer data between representation-theoretic and geometric objects. The most classical instance is the Jordan decomposition for finite groups of Lie type, which identifies a Lusztig series $\mathcal E(G^F,(s))$ with unipotent characters of a dual centralizer (Wang, 2024). Other established uses include the affine-Hecke-algebra map relating the affine Hecke algebra to Lusztig’s asymptotic Hecke algebra (Dawydiak, 2021), the generalized Springer correspondence between relative Weyl-group representations and simple equivariant perverse sheaves (Graham et al., 2020), maps linking Weyl-group conjugacy classes, nilpotent orbits, affine Springer fibers, and strata in reductive groups (Yun, 2020), and several quiver-theoretic and quantum-group realizations of braid symmetries, canonical bases, and Frobenius-type morphisms (Zhao, 2017).

1. Terminological scope

Across the literature represented here, “Lusztig’s map” refers to context-dependent constructions rather than a single invariant. The common pattern is a transfer from a combinatorial or spectral parameter space to a geometric one, or vice versa, together with strong functorial constraints.

Setting	Object called “Lusztig’s map”	Role
Finite groups of Lie type	$L_s$ or $J_s^{\mathrm{can}}$	Jordan decomposition of characters
Affine Hecke algebras	$\phi$	Relates $\mathbf H$ and the asymptotic Hecke algebra $J$
Generalized Springer theory	$\mathrm{Irr}(W_G(L)) \leftrightarrow$ constituents of $A_c$	Classifies simple equivariant perverse sheaves
Reductive groups and affine Springer theory	$\phi$, $\Psi$, $L_s$0, $L_s$1	Connect conjugacy classes, nilpotent orbits, and strata
Quantum groups and quivers	$L_s$2, $L_s$3, $L_s$4	Realize braid symmetries, highest-weight modules, and basis correspondences

A recurring misconception is that these usages are interchangeable. The sources instead exhibit a controlled polysemy: the same label persists because each construction mediates between Lusztig-style cell data, Springer-type geometry, or canonical-basis structures, but the actual domains and codomains differ substantially (Carnovale, 2019).

2. Jordan decomposition for finite groups of Lie type

For a connected reductive algebraic group $L_s$5 over $L_s$6 with Frobenius $L_s$7, Lusztig’s Jordan decomposition partitions $L_s$8 into Lusztig series indexed by semisimple classes in the dual finite group $L_s$9. If $J_s^{\mathrm{can}}$0 is semisimple, the corresponding series is

$J_s^{\mathrm{can}}$1

and Lusztig’s correspondence is a bijection

$J_s^{\mathrm{can}}$2

It is characterized, up to ambiguities when $J_s^{\mathrm{can}}$3 is disconnected, by the Deligne–Lusztig identity

$J_s^{\mathrm{can}}$4

(Wang, 2024).

For finite classical groups, Wang constructs a unique canonical choice

$J_s^{\mathrm{can}}$5

thereby removing the residual ambiguity that remains in disconnected centralizer cases. This construction is canonical for $J_s^{\mathrm{can}}$6 of classical type, including $J_s^{\mathrm{can}}$7, $J_s^{\mathrm{can}}$8 via restriction, $J_s^{\mathrm{can}}$9, $\phi$0, $\phi$1, and $\phi$2. The centralizer $\phi$3 is described through the eigenvalue decomposition of $\phi$4 on the standard representation: the factors attached to $\phi$5 are products of general linear or unitary groups over field extensions, whereas the $\phi$6-eigenspaces contribute orthogonal or symplectic factors. Accordingly, the unipotent side is parametrized by partitions on the $\phi$7 factors and by Lusztig symbols on the orthogonal and symplectic factors (Wang, 2024).

The significance of the canonical choice lies in its functorial compatibilities. First, it is compatible with Harish–Chandra induction: $\phi$8 Second, it is compatible with finite theta correspondence across both orthogonal–symplectic and unitary dual pairs: $\phi$9 up to the explicit partition twists and sign corrections recorded in the orthogonal cases. Wang then uses this canonical map to refine the finite Gan–Gross–Prasad theory and to exhibit a finite-field instance of the relative Langlands duality of Ben-Zvi–Sakellaridis–Venkatesh (Wang, 2024).

This setting is also the clearest illustration that the Jordan decomposition is not automatically canonical. For $\mathbf H$0 and $\mathbf H$1, every irreducible character is uniform, so the Deligne–Lusztig identities determine the map uniquely. In symplectic and orthogonal settings, by contrast, the paper isolates the precise ambiguity and resolves it by a choice compatible simultaneously with parabolic induction and theta correspondence.

3. Affine Hecke algebras and the asymptotic Hecke algebra

In the setting of an extended affine Weyl group $\mathbf H$2, the affine Hecke algebra $\mathbf H$3, and Lusztig’s asymptotic Hecke algebra $\mathbf H$4, “Lusztig’s map” denotes the homomorphism

$\mathbf H$5

defined on the Kazhdan–Lusztig basis by

$\mathbf H$6

After passage to completions and a Goldman involution twist, this yields an embedding

$\mathbf H$7

(Dawydiak, 2021).

This map is organized by Lusztig’s $\mathbf H$8-function and the two-sided cell decomposition of $\mathbf H$9. The asymptotic algebra $J$0 is defined from the leading terms of Kazhdan–Lusztig structure constants, and decomposes as a direct sum $J$1 over two-sided cells. The coefficients $J$2 appearing in the completed Hecke-algebra expansion are controlled by that cell structure: they are rational functions of $J$3, their denominator is independent of $J$4 and depends only on the two-sided cell containing $J$5, and globally the denominator divides a power of the Poincaré polynomial $J$6 of the finite Weyl group. For a distinguished involution $J$7 in the lowest two-sided cell, one has the exact identity

$J$8

(Dawydiak, 2021).

The proof uses Harish–Chandra’s Plancherel formula for $J$9-adic groups. On the analytic side, the map sends $\mathrm{Irr}(W_G(L)) \leftrightarrow$0 to a Harish–Chandra Schwartz function whose Fourier transform is described by explicit densities. For $\mathrm{Irr}(W_G(L)) \leftrightarrow$1, Aubert–Plymen’s formula reduces the argument to contour integrals involving factors of the form

$\mathrm{Irr}(W_G(L)) \leftrightarrow$2

for general $\mathrm{Irr}(W_G(L)) \leftrightarrow$3, Opdam’s $\mathrm{Irr}(W_G(L)) \leftrightarrow$4-function formalism yields analogous denominator control. The resulting denominator theorem is then linked to possible failures of the Kazhdan–Lusztig classification at roots of the Poincaré polynomial (Dawydiak, 2021).

A second consequence is representational. After specializing $\mathrm{Irr}(W_G(L)) \leftrightarrow$5, the induced map from $\mathrm{Irr}(W_G(L)) \leftrightarrow$6 to the Harish–Chandra Schwartz algebra is injective. The paper uses that injectivity to give a criterion for an Iwahori-spherical representation to admit fixed vectors under a larger parahoric subgroup, formulated in terms of its Kazhdan–Lusztig parameter. In this sense, Lusztig’s map is not merely an algebra homomorphism; it is a device that makes two-sided-cell data act faithfully on tempered representation theory.

4. Springer-theoretic and perverse-sheaf meanings

In generalized Springer theory, “Lusztig’s map” denotes the canonical bijection between irreducible representations of a relative Weyl group and simple constituents of a Lusztig sheaf. Given cuspidal data

$\mathrm{Irr}(W_G(L)) \leftrightarrow$7

with $\mathrm{Irr}(W_G(L)) \leftrightarrow$8 a Levi subgroup, $\mathrm{Irr}(W_G(L)) \leftrightarrow$9 a nilpotent $A_c$0-orbit, and $A_c$1 an irreducible $A_c$2-equivariant cuspidal local system, one forms the Lusztig sheaf

$A_c$3

The generalized Springer correspondence is the canonical bijection

$A_c$4

sending $A_c$5 to the unique simple perverse sheaf occurring in $A_c$6 with multiplicity $A_c$7 (Graham et al., 2020).

In type $A_c$8, this correspondence is realized geometrically by a single extended Springer resolution. For $A_c$9, the map

$\phi$0

from the Graham-type extension of the Springer resolution satisfies

$\phi$1

where the summands $\phi$2 are precisely the Lusztig sheaves attached to the cuspidal data determined by central characters $\phi$3. Consequently, one recovers all simple $\phi$4-equivariant perverse sheaves on the nilpotent cone from a single proper map, in the same semismall-decomposition spirit as the classical Springer resolution (Graham et al., 2020).

A related but more representation-theoretic use appears in the symplectic generalized Springer setting. For symplectic partitions with sign data $\phi$5, the generalized Springer correspondence attaches an irreducible Weyl-group representation $\phi$6, while the total even intersection-cohomology representation

$\phi$7

decomposes into a sum of $\phi$8 with multiplicities. The known minimality statement says that $\phi$9 occurs with multiplicity one and is the unique minimal pair under dominance among all constituents. Under the additional hypothesis that $\Psi$0 has only even parts, Waldspurger proves the existence of a unique maximal pair $\Psi$1, again with multiplicity one, and identifies it through a detailed combinatorics of bipartitions, Shoji polynomials, and signature twists (Waldspurger, 2017).

The conceptual point is that in Springer-type contexts, Lusztig’s map is classification-theoretic: it turns singular-support or perverse-sheaf data on nilpotent geometry into explicit Weyl-group representation parameters. The maps are typically canonical only after fixing the appropriate cuspidal block or induction datum.

5. Weyl-group conjugacy, nilpotent orbits, affine Springer fibers, and strata

Another major family of meanings concerns maps between conjugacy classes in Weyl groups, nilpotent orbits, and geometric strata in reductive groups. One version is Lusztig’s map

$\Psi$2

defined originally by intersecting minimal-length representatives of a Weyl-group conjugacy class with double cosets $\Psi$3. In the affine-Springer reinterpretation, one instead defines the minimal reduction type map

$\Psi$4

using shallow affine Springer fibers. The central theorem is that

$\Psi$5

where $\Psi$6 is the Kazhdan–Lusztig map from nilpotent orbits to Weyl-group conjugacy classes, and $\Psi$7 is Lusztig’s canonical section. Moreover, $\Psi$8 is a section of $\Psi$9, so $L_s$00 (Yun, 2020).

This result is proved by analyzing reduction types of affine Springer fibers, the shallow strata in the Chevalley base, and, for classical groups, the “skeleta” fixed by loop tori. The numerical invariant

$L_s$01

controls the dimension of shallow affine Springer fibers, and in the elliptic case it is given by

$L_s$02

The map thereby links Coxeter-theoretic data in $L_s$03 to the geometry of nilpotent orbits and their Springer fibers (Yun, 2020).

A second version acts directly on the group. For a connected reductive group $L_s$04, Lusztig’s map

$L_s$05

assigns to the Jordan decomposition $L_s$06 the truncated induction of the Springer representation of the unipotent part in the connected centralizer. Its fibers are Lusztig’s strata. These strata are locally closed, every stratum is a union of sheets, and the irreducible components of a stratum are precisely the sheets it contains (Carnovale, 2019). In good characteristic, one can also describe the same partition by the Bruhat-theoretic construction $L_s$07, built from minimal-dimension conjugacy classes inside

$L_s$08

for $L_s$09 in a minimal-length Weyl-group conjugacy class $L_s$10; the resulting strata are unions of sheets, and those containing spherical conjugacy classes correspond to unions of classes of involutions in $L_s$11 (Carnovale, 2013).

For spherical unipotent conjugacy classes there is an especially explicit description. If $L_s$12 is the unique Weyl-group element such that $L_s$13 is dense in the spherical unipotent class $L_s$14, then Lusztig’s map $L_s$15 satisfies

$L_s$16

The image of this restricted map consists exactly of those Weyl-group conjugacy classes that have a maximum in Bruhat order; equivalently, those having a unique maximal-length element (Carnovale et al., 2012). This is a sharp instance in which a geometric stratum map becomes a purely Coxeter-theoretic assignment.

Finally, there is an affine analog on two-sided cells of the affine Weyl group. Using singular support on the affine flag variety, one obtains a map from affine two-sided cells to conjugacy classes in the finite Weyl group. The main theorem places this map in a commutative diagram with Lusztig’s bijection between affine two-sided cells and nilpotent orbits of the Langlands dual group, and with the Kazhdan–Lusztig map on the dual side. This furnishes an affine analog of the finite-type triangle “cell $L_s$17 special nilpotent orbit $L_s$18 special representation” (Chua, 2024).

6. Quivers, quantum groups, and categorified forms

In quiver and quantum-group settings, “Lusztig’s map” usually denotes an explicitly functorial realization of braid symmetries, highest-weight modules, or basis correspondences. One basic example is Lusztig’s braid-group automorphism

$L_s$19

given on generators by

$L_s$20

Microlocally, these operators are identified with half-monodromies in vanishing cycles of factorizable sheaves, providing a reformulation of the De Concini–Toledano Laredo conjecture on the Casimir connection (Finkelberg et al., 2014). Geometrically, the same symmetries are realized on the whole quantum group by passing from Lusztig’s perverse-sheaf categories on quiver varieties to a Drinfeld double and constructing a corresponding Hopf algebra isomorphism at the categorical level (Zhao, 2017).

Another cluster of meanings concerns the realization of highest-weight modules and canonical bases by Lusztig sheaves. For framed quivers, one has a canonical $L_s$21-module isomorphism

$L_s$22

sending the class of the constant sheaf to the highest-weight vector $L_s$23. The nonzero simple perverse sheaves in the localized categories form the canonical basis of $L_s$24 (Fang et al., 2023). Closely related to this, the canonical and semicanonical bases are connected by a graph isomorphism

$L_s$25

which identifies the two inductive algorithms used by Lusztig. As a consequence, the transition matrix from the canonical basis to the semicanonical basis is upper triangular with all diagonal entries equal to $L_s$26 (Fang et al., 2022).

Several further quiver-theoretic constructions also carry the name. Lusztig’s iterated convolution diagram for the cyclic quiver produces proper morphisms

$L_s$27

whose geometry controls multigraded multiplicities and the $L_s$28-variable Kostka–Shoji polynomials via global sections of line bundles on the corresponding convolution bundle (Finkelberg et al., 2016). In the mixed equivariant setting for a quiver with automorphism, Lusztig’s induction and restriction functors intertwine with trace maps to give function-level Hall multiplication and comultiplication, yielding Green’s formula for every finite-dimensional hereditary algebra over a finite field (Fang et al., 2024). At a prime root of unity, Lusztig’s quantum Frobenius for $L_s$29,

$L_s$30

admits a categorification through $L_s$31-DG techniques and a thickening functor between thin and thick graphical calculi (Qi, 2016).

In these quantum and quiver settings, the term no longer denotes a single bijection between two discrete parameter sets. It instead marks a family of functorial transports—between monodromy and braid operators, Grothendieck groups and integrable modules, perverse sheaves and basis vectors, or quantum groups at roots of unity and their classical limits. The unifying feature is structural exactness: these maps preserve induction, restriction, duality, or braid relations in a form rigid enough to serve as a replacement for direct calculation.