Higher Lusztig Induction

• Higher Lusztig Induction is a family of geometric and homological constructions that generalize classical Deligne–Lusztig induction to finite local rings and p-adic groups.
• It provides explicit inner product formulas and algebraisation results, including Mackey-type identities and induced models at both even and odd congruence levels.
• Derived and categorical refinements, such as quantum group and D-module approaches, enhance its applications in representation theory and character sheaf analysis.

Higher Lusztig induction denotes a family of extensions of classical Deligne–Lusztig and Lusztig induction beyond the setting of finite reductive groups over finite fields. In the literature surveyed here, the phrase is used most directly for cohomological constructions attached to higher Deligne–Lusztig varieties over finite local rings and to parahoric quotients of $p$-adic groups, where virtual representations such as $R_{T,U,b}^{\theta}$ and $R_{L,P,r}^G(\psi)$ are defined from compactly supported $\ell$-adic cohomology (Chen, 2017, Yu, 16 Aug 2025). Closely related usages treat derived and categorical refinements of Lusztig induction, including derived induction equivalences for quantum groups and algebraic groups and torus-level $D$-module kernels for parabolic induction and restriction (Hodge et al., 2016, Ginzburg, 2021). This suggests that “higher” refers either to passage to higher congruence level, to positive $p$-adic depth, or to derived and categorical enhancement, while the unifying theme is the replacement of the classical Deligne–Lusztig correspondence by more elaborate geometric or homological machinery.

1. Terminological scope and basic framework

A common source of confusion is that the expression does not denote a single universally fixed construction across all subliteratures. The most direct uses concern higher Deligne–Lusztig theory for groups over finite quotients of discrete valuation rings and its parahoric or deep-level analogues for $p$-adic groups, but several papers also use the phrase to describe derived, sheaf-theoretic, or categorical realizations of Lusztig-type induction (Yu, 16 Aug 2025, Ginzburg, 2021).

Setting	Basic object	Representative reference
Finite local rings	$$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$	(Chen, 2017)
Parahoric quotients	$$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$	(Yu, 16 Aug 2025)
Deep-level tame $p$-adic theory	$R_{T,U,b}^{\theta}$0 and $R_{T,U,b}^{\theta}$1	(Ivanov et al., 11 Jan 2026)
Derived and categorical refinements	$R_{T,U,b}^{\theta}$2, $R_{T,U,b}^{\theta}$3, $R_{T,U,b}^{\theta}$4	(Hodge et al., 2016, Ginzburg, 2021)

In the parahoric formulation, higher Lusztig induction is explicitly described as constructing “a broad class of virtual smooth representations of parahoric subgroups in a $R_{T,U,b}^{\theta}$5-adic group,” and as a natural generalization of classical Lusztig induction to the $R_{T,U,b}^{\theta}$6-adic setting (Yu, 16 Aug 2025). In the finite-local-ring formulation, the same philosophy appears through Greenberg realizations of reductive group schemes over $R_{T,U,b}^{\theta}$7, with $R_{T,U,b}^{\theta}$8, Lang preimages, and cohomological virtual representations (Chen, 2017).

2. Higher Deligne–Lusztig induction over finite local rings

One fundamental realization of higher Lusztig induction begins with a complete discrete valuation ring $R_{T,U,b}^{\theta}$9, uniformizer $R_{L,P,r}^G(\psi)$0, quotient rings $R_{L,P,r}^G(\psi)$1, and a connected reductive group scheme over $R_{L,P,r}^G(\psi)$2. After Greenberg realization, one obtains an algebraic group $R_{L,P,r}^G(\psi)$3 over $R_{L,P,r}^G(\psi)$4 equipped with Frobenius $R_{L,P,r}^G(\psi)$5 such that

$R_{L,P,r}^G(\psi)$6

Fixing an $R_{L,P,r}^G(\psi)$7-stable maximal torus $R_{L,P,r}^G(\psi)$8, a Borel with unipotent radical $R_{L,P,r}^G(\psi)$9, and the Lang map $\ell$0, one defines for $\ell$1 the Deligne–Lusztig variety at page $\ell$2,

$\ell$3

and the associated virtual representation

$\ell$4

This family interpolates between higher Deligne–Lusztig and Gérardin-type constructions (Chen, 2017).

The endpoints of the family recover previously known objects. The paper states that when $\ell$5, $\ell$6 coincides with the higher Deligne–Lusztig representation introduced by Lusztig, and when $\ell$7 is even and $\ell$8, it coincides with Gérardin’s representation. It also notes a slight indexing inconsistency between the introduction and later notation, while emphasizing that the mathematical substance is clear: one endpoint of the family is the standard higher Deligne–Lusztig construction, and every intermediate page-$\ell$9 representation is compared against it (Chen, 2017).

The key theorem is an inner product formula. If $D$0 is regular, then

$D$1

and if $D$2 is moreover in general position, then the inner product is $D$3. Here regularity is the usual higher-level condition requiring $D$4 to be non-trivial on every norm subgroup

$D$5

for roots $D$6, and general position means precisely that the Weyl-group stabilizer is trivial (Chen, 2017).

This formula gives a precise computational relation between generalized page-$D$7 constructions and higher Lusztig induction. It does not identify $D$8 with $D$9, nor does it prove irreducibility, but it shows that under regularity the overlap is controlled exactly by a Weyl-stabilizer cardinality, and in the generic case it is minimal and nonzero. A plausible implication is that higher Lusztig induction is better viewed as a family of geometrically natural Lang-preimage constructions rather than a single isolated variety.

3. Algebraisation and induced models at even and odd level

A second major strand identifies higher Deligne–Lusztig representations with algebraically induced representations from congruence-level subgroups. At even level $p$0, one defines the arithmetic radical

$p$1

The paper proves that

$p$2

where $p$3 is the trivial extension of $p$4 to $p$5, and for generic $p$6 one obtains

$p$7

Here generic means regular, in general position, and satisfying the stabilizer condition

$p$8

The paper presents this as the even-level solution to Lusztig’s algebraisation problem in the generic case (Chen et al., 2016).

The odd-level case is substantially subtler. For arbitrary level $p$9, with

$p$0

the all-level algebraisation theorem states that for strongly generic $p$1,

$p$2

At odd levels the inducing representation is no longer an inflated torus character. One first constructs a Heisenberg lift $p$3 on $p$4, characterized by

$p$5

and then extends it to $p$6 as $p$7. The same paper also proves an induction formula

$p$8

and gives a new proof of the regular semisimple character formula (Chen et al., 2023).

The level dependence clarifies one of the central features of higher Lusztig induction. At even level, algebraisation is controlled by a torus character on a congruence extension of $p$9. At odd level, a genuinely non-abelian Heisenberg step intervenes. For $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$0, the same paper proves that regular, generic, and strongly generic are equivalent, so the stabilizer condition becomes particularly transparent in type $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$1 (Chen et al., 2023).

4. Parahoric and deep-level $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$2-adic forms

In the parahoric setting, higher Lusztig induction is defined for a $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$3-rational Levi subgroup $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$4, a parabolic $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$5, and the higher Deligne–Lusztig variety

$$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$6

The associated induction functor is

$$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$7

and when $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$8 it recovers classical Lusztig induction. The main theorem in the generic case is a Mackey-type scalar-product formula: $$R_{T,U,b}^{\theta}=H_c^*(L^{-1}(FU^{r-b,b}),\overline{\mathbb Q}_\ell)_\theta$$9 for $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$0-generic constituents $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$1 arising from a generic toral representation of $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$2. In the elliptic-torus case this yields a decomposition

$$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$3

where the $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$4 are pairwise non-isomorphic irreducible representations obtained by successive higher Lusztig inductions along a Howe factorization (Yu, 16 Aug 2025).

A complementary development gives an explicit decomposition of elliptic higher Deligne–Lusztig representations into Yu-type building blocks. If $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$5 is a Howe factorization, then previous work had shown

$$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$6

The refinement identifies the geometric $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$7 with the classical Weil–Heisenberg representation $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$8 up to the Fintzen–Kaletha–Spice quadratic character and, under a mild condition on $$R_{L,P,r}^G(\psi)=H_c^*(X^G_{L,P,r},\overline{\mathbb Q}_\ell)\otimes_{L_r^F}\psi$$9, no further discrepancy: $p$0 hence

$p$1

Under the same mild condition on $p$2, each unramified Yu type appears in the cohomology of higher Deligne–Lusztig varieties, and each unramified Kaletha regular supercuspidal representation is the compact induction of a specified higher Deligne–Lusztig representation up to a sign (Liu et al., 15 Jun 2025).

The tamely ramified deep-level theory extends this picture beyond the unramified case. For a tame elliptic torus $p$3 and a character $p$4 admitting a Howe factorization, one constructs a deep-level Deligne–Lusztig variety

$p$5

and the virtual representation

$p$6

Under a largeness condition on $p$7, the comparison theorem states

$p$8

and if $p$9, then likewise

$R_{T,U,b}^{\theta}$00

As a consequence, each regular irreducible supercuspidal is the compact induction of a deep-level Deligne–Lusztig representation, and more generally each irreducible supercuspidal is a direct summand of the compact induction of the cohomology of a deep-level Deligne–Lusztig variety (Ivanov et al., 11 Jan 2026).

Another $R_{T,U,b}^{\theta}$01-adic realization appears for division algebras. For the $R_{T,U,b}^{\theta}$02-adic Deligne–Lusztig ind-scheme $R_{T,U,b}^{\theta}$03 attached to a division algebra $R_{T,U,b}^{\theta}$04 of invariant $R_{T,U,b}^{\theta}$05, and a primitive character $R_{T,U,b}^{\theta}$06 of level $R_{T,U,b}^{\theta}$07, the paper proves

$R_{T,U,b}^{\theta}$08

where $R_{T,U,b}^{\theta}$09 is the irreducible smooth representation of $R_{T,U,b}^{\theta}$10 attached to $R_{T,U,b}^{\theta}$11 by local Langlands and Jacquet–Langlands. This is a successful $R_{T,U,b}^{\theta}$12-adic instance of the same general philosophy (Chan, 2015).

5. Derived and categorical reformulations

In a different direction, higher Lusztig induction appears as a derived or categorical enhancement of classical induction. One influential realization is the ABG induction theorem and its modular analogue. In the quantum case, for a Lusztig quantum enveloping algebra $R_{T,U,b}^{\theta}$13 at a root of unity and its negative Borel part $R_{T,U,b}^{\theta}$14, the theorem states

$R_{T,U,b}^{\theta}$15

For a semisimple, simply connected algebraic group $R_{T,U,b}^{\theta}$16 in characteristic $R_{T,U,b}^{\theta}$17, the modular analogue is

$R_{T,U,b}^{\theta}$18

Here “higher” is homological: the right derived induction functor recovers the entire bounded derived category of the principal block from a triangulated subcategory on the Borel side. The same work also notes truncation refinements for $R_{T,U,b}^{\theta}$19 (Hodge et al., 2016).

A second realization occurs in torus-level $R_{T,U,b}^{\theta}$20-module theory. For a connected and simply-connected complex semisimple group $R_{T,U,b}^{\theta}$21 with maximal torus $R_{T,U,b}^{\theta}$22, parabolic induction and restriction are made completely explicit in the case $R_{T,U,b}^{\theta}$23. The functors are

$R_{T,U,b}^{\theta}$24

where

$R_{T,U,b}^{\theta}$25

The correspondence kernel is the Harish-Chandra $R_{T,U,b}^{\theta}$26-module

$R_{T,U,b}^{\theta}$27

and the paper proves

$R_{T,U,b}^{\theta}$28

It also proves that the $R_{T,U,b}^{\theta}$29-bimodule $R_{T,U,b}^{\theta}$30 is flat as a right $R_{T,U,b}^{\theta}$31-module, hence $R_{T,U,b}^{\theta}$32 is $R_{T,U,b}^{\theta}$33-exact, while

$R_{T,U,b}^{\theta}$34

This is a highly explicit derived and kernel-theoretic model of Lusztig-type induction in the torus case (Ginzburg, 2021).

A third categorical usage appears in quiver geometry. There, Lusztig’s induction and restriction functors on equivariant derived categories realize multiplication and comultiplication in the Hall-algebra model of $R_{T,U,b}^{\theta}$35, while derivation functors are special restriction functors. The basic induction is

$R_{T,U,b}^{\theta}$36

and the paper proves derived refinements of the induction–restriction compatibility in the form of distinguished triangles such as

$R_{T,U,b}^{\theta}$37

This strand does not define a new functor called higher Lusztig induction, but it treats the “higher” content as the passage from algebraic identities to functorial and triangulated statements in derived categories (Zhao, 2022).

6. Structural consequences, comparison principles, and recurring misconceptions

Several adjacent results clarify what higher Lusztig induction does and does not provide. On the Fourier-analytic side, one paper proves that for a semisimple Lie algebra $R_{T,U,b}^{\theta}$38 over $R_{T,U,b}^{\theta}$39, if $R_{T,U,b}^{\theta}$40, then

$R_{T,U,b}^{\theta}$41

and deduces Letellier’s conjectural compatibility of Fourier transform with Deligne–Lusztig induction,

$R_{T,U,b}^{\theta}$42

This does not define a higher induction functor, but it shows that induction interacts with Fourier transform by explicit scalar laws controlled by the Weil index and the ranks of the source and target groups (Liu et al., 23 Apr 2026).

On the character-sheaf side, Lusztig induction for arbitrary $R_{T,U,b}^{\theta}$43-stable Levi subgroups is related to parabolic induction of character sheaves and to induction on relative Weyl-group cosets. Under the weak Lusztig conjecture and a compatibility property between character sheaves and Deligne–Lusztig induction, the paper proves that if

$R_{T,U,b}^{\theta}$44

then

$R_{T,U,b}^{\theta}$45

It also establishes the comparison formula

$R_{T,U,b}^{\theta}$46

which places Lusztig induction inside a relative Weyl-group induction formalism (Taylor et al., 2018).

On the Hecke-theoretic side, Lusztig induction operators for finite $R_{T,U,b}^{\theta}$47 and $R_{T,U,b}^{\theta}$48 are identified with Heisenberg operators on Fock space and, blockwise, with parabolic induction in cyclotomic rational double affine Hecke algebras. The core equalities are

$R_{T,U,b}^{\theta}$49

on the finite-group side and

$R_{T,U,b}^{\theta}$50

under the blockwise correspondence with CRDAHA category $R_{T,U,b}^{\theta}$51. This gives a categorical and Hecke-theoretic lift of Lusztig induction on unipotent blocks (Srinivasan, 2014).

These comparison theorems also delimit the subject. A recurring misconception is that higher Lusztig induction always supplies a single universal functor together with a full Mackey formula, an explicit irreducible decomposition, and an exact equality with algebraically induced models. The literature is more stratified. The page-$R_{T,U,b}^{\theta}$52 theory gives an inner product theorem rather than equality (Chen, 2017). The parahoric theory proves a generic scalar-product Mackey formula rather than a full operator identity (Yu, 16 Aug 2025). The all-level algebraisation theorem identifies higher Deligne–Lusztig representations with induced models for strongly generic characters, but the odd-level proof requires Heisenberg extensions and explicit sign control (Chen et al., 2023). A plausible summary is that higher Lusztig induction is best understood not as a single theorem, but as a research program whose central achievements are comparison formulas, algebraisation statements, derived realizations, and explicit decompositions in increasingly general geometric settings.