Higher Deligne–Lusztig Reps

• Higher Deligne–Lusztig representations are cohomologically defined virtual representations extending classical induction methods to settings like finite rings, parahoric quotients, and semi-infinite geometries.
• They provide a geometric framework for constructing irreducible representations, establishing precise character formulas, and linking algebraic induction with local Langlands phenomena.
• Applications include computing Green functions, analyzing orbit structures, and unifying geometric and algebraic methods in the study of p-adic and loop group representations.

Higher Deligne–Lusztig representations are cohomologically defined virtual representations attached to reductive groups over finite quotients of discrete valuation rings, parahoric quotients of $p$-adic groups, and related semi-infinite or loop-theoretic spaces. They generalize classical Deligne–Lusztig induction from finite reductive groups to settings with positive depth, congruence filtration, and affine or semi-infinite geometry. In the finite-ring model, one starts from a connected reductive group scheme $\mathbb{G}$ over a discrete valuation ring $\mathcal O$, passes to the level-$r$ quotient $\mathcal O_r=\mathcal O/\varpi^r$, and forms the Lang-preimage variety $L^{-1}(FU)$ or its mixed-level variants; for a character $\theta\in \mathrm{Irr}(T^F)$, the associated virtual representation is

$$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$

In parahoric and deep-level settings the same pattern persists, but $G_r$ is replaced by Moy–Prasad quotients or positive-loop group schemes, and the resulting cohomology realizes positive-depth representation-theoretic constructions. Across these settings, higher Deligne–Lusztig theory serves as a geometric mechanism for producing irreducible representations, establishing character formulas, and relating geometric constructions to algebraic inductions, Yu types, and local Langlands phenomena (Chen et al., 2016, Nie, 2024, Ivanov et al., 11 Jan 2026).

1. Finite-ring foundations and the higher Lang construction

The finite-ring theory begins with a connected reductive group scheme $\mathbb G$ over a complete discrete valuation ring $\mathbb{G}$0 with residue field $\mathbb{G}$1, and the level quotient $\mathbb{G}$2. After passage through the Greenberg functor, one obtains an algebraic group $\mathbb{G}$3 over $\mathbb{G}$4 with Frobenius $\mathbb{G}$5, together with reduction morphisms

$\mathbb{G}$6

whose kernels $\mathbb{G}$7 form the congruence filtration. For a maximal torus $\mathbb{G}$8 and a Borel $\mathbb{G}$9, the basic higher Deligne–Lusztig variety is the Lang fiber $\mathcal O$0, where $\mathcal O$1, and the associated virtual representation is the alternating compactly supported $\mathcal O$2-adic cohomology in the $\mathcal O$3-isotypic part (Chen et al., 2016, Chen et al., 2023).

A parallel formulation uses mixed-level subgroups

$\mathcal O$4

and defines

$\mathcal O$5

This interpolates between the standard higher Deligne–Lusztig representation $\mathcal O$6 and other Deligne–Lusztig type constructions relevant to algebraisation and Gérardin-type representations (Chen, 2018, Chen, 2017).

Genericity conditions are central. The finite-ring literature distinguishes general position, regularity, genericity, and strong genericity. Regularity is expressed by nontriviality on norm images from root subtori, general position by trivial stabilizer in the finite Weyl group, and strong genericity by an additional stabilizer condition on the inflated character $\mathcal O$7 along congruence subgroups. Under regularity and general position, higher Deligne–Lusztig representations are irreducible up to sign; this extends the classical irreducibility criterion to finite local rings (Chen et al., 2023, Chen et al., 2016).

The same formalism admits combinatorial and flag-theoretic enlargements. For $\mathcal O$8 and $\mathcal O$9, admissible flags in $r$0 give a family of virtual representations

$r$1

that contains ordinary higher Deligne–Lusztig representations as special cases and, in the stated ranges, affords all nilpotent orbit representations, all regular orbit representations for $r$2, and invariant Lie algebra characters when $r$3 (Chen, 2019).

2. Character formulas, Green functions, and inner products

A major structural advance in higher Deligne–Lusztig theory is the extension of the Deligne–Lusztig character formula from finite fields to finite local rings. For $r$4 with Jordan decomposition, the higher character $r$5 is expressed as a sum over conjugacy data $r$6 with $r$7, together with a second sum over $r$8, and a trace on compactly supported cohomology of a centralizer-intersected Deligne–Lusztig variety. This motivates a two-variable Green function

$r$9

which depends simultaneously on a unipotent element $\mathcal O_r=\mathcal O/\varpi^r$0 and a torus congruence parameter $\mathcal O_r=\mathcal O/\varpi^r$1 (Chen, 2018).

These Green functions satisfy the analogues of classical summation identities. In particular,

$\mathcal O_r=\mathcal O/\varpi^r$2

and the averaged Green function

$\mathcal O_r=\mathcal O/\varpi^r$3

is $\mathcal O_r=\mathcal O/\varpi^r$4-valued on unipotent classes. The dependence on $\mathcal O_r=\mathcal O/\varpi^r$5 is a genuinely higher-level phenomenon absent at level $\mathcal O_r=\mathcal O/\varpi^r$6 (Chen, 2018).

At regular semisimple elements, the character formula simplifies sharply. If $\mathcal O_r=\mathcal O/\varpi^r$7 is regular semisimple and its conjugacy class meets $\mathcal O_r=\mathcal O/\varpi^r$8, then

$\mathcal O_r=\mathcal O/\varpi^r$9

independent of $L^{-1}(FU)$0 and $L^{-1}(FU)$1. In the finite-ring setting of Chen–Stasinski, the analogous formula is

$L^{-1}(FU)$2

for $L^{-1}(FU)$3 regular semisimple and $L^{-1}(FU)$4 strongly generic. The proof uses Broué’s generalized bimodule induction formula together with a centralizer analysis of the fixed-point subvariety (Chen, 2018, Chen et al., 2023).

Inner-product formulas form the orthogonality theory of higher Deligne–Lusztig representations. Beyond the classical level-one formula, one now has higher-level analogues for several families. For the generalized page-$L^{-1}(FU)$5 varieties,

$L^{-1}(FU)$6

for regular $L^{-1}(FU)$7. For Coxeter-type deep-level Deligne–Lusztig schemes over parahoric quotients, the same orthogonality relation holds under the explicit bound $L^{-1}(FU)$8, with $L^{-1}(FU)$9 depending only on the root system, and without any regularity assumption on $\theta\in \mathrm{Irr}(T^F)$0. This includes the unipotent case $\theta\in \mathrm{Irr}(T^F)$1 and extends previous regular-character results (Chen, 2017, Dudas et al., 2020).

3. Algebraisation and induced-model descriptions

A central problem, raised by Lusztig, asks whether the cohomological higher Deligne–Lusztig representations coincide with explicitly induced algebraic representations. At even levels $\theta\in \mathrm{Irr}(T^F)$2, this was resolved by identifying the geometric representation with induction from the “arithmetic radical”

$\theta\in \mathrm{Irr}(T^F)$3

a commutative $\theta\in \mathrm{Irr}(T^F)$4-rational subgroup normalized by $\theta\in \mathrm{Irr}(T^F)$5. In the generic case,

$\theta\in \mathrm{Irr}(T^F)$6

and consequently

$\theta\in \mathrm{Irr}(T^F)$7

This answered the even-level algebraisation problem and yielded applications such as the verification of Letellier’s conjecture for $\theta\in \mathrm{Irr}(T^F)$8 and $\theta\in \mathrm{Irr}(T^F)$9 (Chen et al., 2016).

The odd-level case requires additional structure. For $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$0, Chen–Stasinski proved a full algebraisation theorem. If $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$1 is strongly generic, then

$$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$2

where $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$3, $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$4, and $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$5 is characterized by trace formulas on $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$6 and on $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$7. In odd level, a Heisenberg representation $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$8 of $$R_{T,U}^{\theta}:=\sum_i(-1)^i H_c^i\!\big(L^{-1}(FU),\overline{\mathbb Q_\ell}\big)_\theta.$$9 intervenes, with

$G_r$0

and $G_r$1 admits extensions to $G_r$2. For $G_r$3, the sign is explicitly determined by

$G_r$4

This completes algebraisation at all levels $G_r$5 (Chen et al., 2023).

An induction formula underlies the proof: $G_r$6 where

$G_r$7

Thus the geometric representation is induced from an intermediate cohomology module on a smaller subgroup. This induction description is also the vehicle for the new proof of the regular-semisimple character formula and for the explicit determination of odd-level extensions (Chen et al., 2023).

The relation to Gérardin’s algebraic constructions is also clarified. Under the hypotheses stated in the paper, and for $G_r$8, the signed higher Deligne–Lusztig representation

$G_r$9

is isomorphic to Gérardin’s representation whenever the latter is defined. This resolves Lusztig’s problem in the generic regime and unifies geometric and algebraic constructions across even and odd levels (Chen et al., 2023).

4. Orbits, dimensions, stability, and low-rank phenomena

Higher Deligne–Lusztig representations interact naturally with orbit theory on the last congruence layer. Under assumptions guaranteeing a nondegenerate invariant bilinear form $\mathbb G$0 on $\mathbb G$1, one obtains a map

$\mathbb G$2

to adjoint orbits in the Lie algebra. For regular $\mathbb G$3, at least one irreducible constituent of $\mathbb G$4 has regular semisimple orbit, and if $\mathbb G$5 is generic then $\mathbb G$6 is regular and semisimple. In $\mathbb G$7, regular semisimple orbit representations are precisely those of the form $\mathbb G$8 with $\mathbb G$9 regular (Chen et al., 2023).

Dimension formulas are now known in several regimes. For strongly generic $\mathbb{G}$00 and arbitrary $\mathbb{G}$01,

$\mathbb{G}$02

For $\mathbb{G}$03,

$\mathbb{G}$04

These formulas were previously unknown at odd levels and align with the classical Steinberg-based formula when $\mathbb{G}$05 (Chen et al., 2023).

A distinct phenomenon appears for Coxeter tori. If $\mathbb{G}$06 is Coxeter and $\mathbb{G}$07 or $\mathbb{G}$08, then the Coxeter unipotent representation stabilizes with the level: $\mathbb{G}$09 More generally, if $\mathbb{G}$10 is the trivial inflation of a lower-level character $\mathbb{G}$11, then

$\mathbb{G}$12

This shows that, in the Coxeter unipotent case, higher-level geometry does not produce new representations beyond trivial inflation from lower levels (Chen, 2024).

Low-rank calculations make the picture explicit. For $\mathbb{G}$13 with Coxeter $\mathbb{G}$14, if $\mathbb{G}$15 is regular then $\mathbb{G}$16 is irreducible of dimension $\mathbb{G}$17. If $\mathbb{G}$18 is nonregular, the representation reduces to lower conductor level $\mathbb{G}$19, and the dimensions that occur are exactly

$\mathbb{G}$20

For $\mathbb{G}$21, analogous decompositions hold with stated parity and quadratic-character exceptions (Chen, 2024).

Finally, non-nilpotency criteria distinguish higher Deligne–Lusztig representations from nilpotent orbit representations. If $\mathbb{G}$22 is regular and in general position, and is nontrivial on the biggest reduction kernel of the center, then $\mathbb{G}$23, up to sign, is not nilpotent. This is proved by comparing it against the induced representation $\mathbb{G}$24, whose irreducible constituents are precisely the nilpotent representations (Chen, 2019).

5. Parahoric, deep-level, and Yu-type decompositions

In the parahoric setting, higher Deligne–Lusztig representations become virtual smooth representations of $\mathbb{G}$25 or of the ambient $\mathbb{G}$26-adic group via inflation. For $\mathbb{G}$27, the higher or parahoric Deligne–Lusztig variety is

$\mathbb{G}$28

and

$\mathbb{G}$29

For elliptic $\mathbb{G}$30, Chan’s inner-product formula gives

$\mathbb{G}$31

and $\mathbb{G}$32 is independent of the choice of Borel containing $\mathbb{G}$33 (Nie, 2024).

A decisive structural theorem proves that elliptic higher Deligne–Lusztig representations decompose in the same pattern as Yu’s tame supercuspidal construction. Under the standing prime condition

$\mathbb{G}$34

and with a Howe factorization $\mathbb{G}$35, one introduces a geometric Heisenberg-type module

$\mathbb{G}$36

Then

$\mathbb{G}$37

where $\mathbb{G}$38 runs over irreducible constituents of the classical depth-zero Deligne–Lusztig representation $\mathbb{G}$39, and each summand is irreducible and pairwise non-isomorphic. The module $\mathbb{G}$40 is the geometric analogue of Yu’s Weil–Heisenberg representation (Nie, 2024).

This decomposition was later made explicit. The geometric Weil–Heisenberg representation $\mathbb{G}$41 differs from Yu’s analytic $\mathbb{G}$42 by a character $\mathbb{G}$43, and under a mild condition on $\mathbb{G}$44 one has

$\mathbb{G}$45

where $\mathbb{G}$46 is the quadratic character of Fintzen–Kaletha–Spice. Consequently,

$\mathbb{G}$47

giving an explicit irreducible decomposition of elliptic higher Deligne–Lusztig representations (Liu et al., 15 Jun 2025).

Deep-level Deligne–Lusztig induction for tamely ramified tori extends this framework beyond unramified tori. One constructs a perfect affine scheme

$\mathbb{G}$48

attached to a tamely ramified torus $\mathbb{G}$49, a generic datum $\mathbb{G}$50, and a Lagrangian in a Heisenberg extension. Its cohomology defines a virtual representation

$\mathbb{G}$51

Under the stated largeness assumptions on $\mathbb{G}$52, one proves

$\mathbb{G}$53

where $\mathbb{G}$54 is the model representation built from the Weil–Heisenberg representation and a classical Deligne–Lusztig factor. As an application, 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).

A complementary generic Mackey formula for higher Lusztig functors was also established. If $\mathbb{G}$55 are $\mathbb{G}$56-generic constituents of $\mathbb{G}$57, then

$\mathbb{G}$58

For elliptic $\mathbb{G}$59, this yields irreducibility and a decomposition theorem compatible with Howe factorization (Yu, 16 Aug 2025).

6. Semi-infinite, loop, and division-algebra realizations

Beyond finite rings and parahorics, higher Deligne–Lusztig theory enters a genuinely $\mathbb{G}$60-adic regime through semi-infinite and loop Deligne–Lusztig varieties. For division algebras, Lusztig’s semi-infinite Deligne–Lusztig variety $\mathbb{G}$61 is an ind-scheme whose finite-type approximations $\mathbb{G}$62 carry commuting actions of torus and unipotent groups. For a character $\mathbb{G}$63 of the finite torus $\mathbb{G}$64, the alternating sum

$\mathbb{G}$65

is irreducible, and in fact $\mathbb{G}$66 is nonzero in exactly one degree $\mathbb{G}$67. Moreover, $\mathbb{G}$68 is maximal in the sense that Frobenius acts on $\mathbb{G}$69 by $\mathbb{G}$70, so the cohomology is pure of weight $\mathbb{G}$71 (Chan, 2016).

For the full semi-infinite variety attached to a division algebra $\mathbb{G}$72, the homology $\mathbb{G}$73 attached to a smooth character $\mathbb{G}$74 is concentrated in one degree and yields an irreducible $\mathbb{G}$75-representation when $\mathbb{G}$76 has trivial Galois stabilizer. The trace at very regular elements is

$\mathbb{G}$77

and the resulting correspondence matches the local Langlands and Jacquet–Langlands correspondences (Chan, 2016, Chan, 2015).

For $\mathbb{G}$78, this program was worked out concretely for the quaternion division algebra: if $\mathbb{G}$79 is primitive of level $\mathbb{G}$80, then

$\mathbb{G}$81

where $\mathbb{G}$82 is the Jacquet–Langlands transfer of the $\mathbb{G}$83-supercuspidal attached to $\mathbb{G}$84 by local Langlands (Chan, 2014).

Loop Deligne–Lusztig varieties of Coxeter type provide a parallel realization for inner forms of $\mathbb{G}$85. For an unramified elliptic maximal torus $\mathbb{G}$86, the loop Deligne–Lusztig variety $\mathbb{G}$87 has truncated models $\mathbb{G}$88, and for $\mathbb{G}$89 in general position one obtains

$\mathbb{G}$90

where $\mathbb{G}$91 is the Jacquet–Langlands transfer of the local Langlands representation attached to

$\mathbb{G}$92

This gives a purely local geometric realization of many supercuspidals whose parameter factors through an unramified elliptic torus (Chan et al., 2019).

A related phenomenon appears for $\mathbb{G}$93. There, the semi-infinite Deligne–Lusztig variety $\mathbb{G}$94 is identified with an inverse limit of higher-level affine Deligne–Lusztig varieties: $\mathbb{G}$95 Moreover, a distinguished component admits a product decomposition, up to perfection,

$\mathbb{G}$96

so its representation-theoretic content is governed by a classical Deligne–Lusztig factor. The finite-type varieties $\mathbb{G}$97 studied by Chan and Ivanov then become components of the semi-infinite Deligne–Lusztig space at infinite level, interpreting their cohomology as a realization of Lusztig’s conjectural $\mathbb{G}$98-adic construction for $\mathbb{G}$99 (Takamatsu, 2023).

Finally, recent categorical work reinterprets Deligne–Lusztig theory itself through higher categorical trace. In that framework,

$\mathcal O$00

and Deligne–Lusztig induction arises as a trace of a Frobenius-twisted induction 1-morphism. This suggests a categorical extension of higher Deligne–Lusztig theory to loop groups and isocrystal stacks, though the loop case requires additional stack-theoretic input because Lang’s theorem is no longer available (Gaitsgory et al., 27 Mar 2026).