Non-Abelian Exotic Gauss Sums Theory

• Non-Abelian exotic Gauss sums are normalized finite-field matrix exponential sums attached to irreducible representations of GL_c(F_q), combining determinant and trace phases via Shintani correspondence.
• They reduce to products of abelian exotic Gauss sums through regular-character data and the Hasse–Davenport identities, simplifying complex computations.
• This framework unifies exponential sum analysis by connecting Hall–Littlewood polynomials, exotic Kloosterman sheaves, and Bessel–Speh functions, with implications in finite-field representation theory.

Searching arXiv for the cited papers and closely related work to ground the article. Non-Abelian exotic Gauss sums are normalized finite-field matrix exponential sums attached to an irreducible representation of $GL_c(F_q)$ and to multiplicative characters on finite extensions or, more generally, on étale $F_q$-algebras. They are defined by combining determinant and trace phases on $GL_c(F_{q^k})$ with Shintani’s correspondence between Frobenius-twisted conjugacy classes in $GL_c(F_{q^k})$ and ordinary conjugacy classes in $GL_c(F_q)$. In the sense of Deligne and Katz, the adjective “exotic” refers to sums over commutative étale $F_q$-algebras rather than only over finite fields; the matrix theory developed in "On exotic matrix exponential sums and Bessel-Speh functions" extends this viewpoint to a genuinely non-abelian $GL_c$-setting and connects it with Hall–Littlewood theory, Katz’s exotic Kloosterman sheaf, and special values of Bessel functions attached to Speh representations (Zelingher, 8 Jul 2025, Wan et al., 4 Jun 2026).

1. Definition, normalization, and Shintani correspondence

Let $F=F_q$ be a finite field, let $\psi:F\to \mathbf{C}$ be a fixed non-trivial additive character, and for $k\ge 1$ write $F_q$0 for the degree-$F_q$1 extension inside a fixed algebraic closure. The canonical additive character on $F_q$2 is

$F_q$3

For $F_q$4, the Shintani norm is

$F_q$5

Although $F_q$6 need not lie in $F_q$7, it is $F_q$8-conjugate to an element of $F_q$9; the associated conjugacy class in $GL_c(F_{q^k})$0 is denoted $GL_c(F_{q^k})$1. Shintani proved that

$GL_c(F_{q^k})$2

is a bijection from Frobenius-twisted conjugacy classes of $GL_c(F_{q^k})$3 to conjugacy classes of $GL_c(F_{q^k})$4 (Zelingher, 8 Jul 2025).

For an irreducible representation $GL_c(F_{q^k})$5 of $GL_c(F_{q^k})$6 and a multiplicative character $GL_c(F_{q^k})$7, the non-Abelian exotic Gauss sum is

$GL_c(F_{q^k})$8

Since this is a $GL_c(F_{q^k})$9-equivariant operator on $GL_c(F_{q^k})$0, Schur’s lemma yields a scalar, denoted by the same symbol, such that

$GL_c(F_{q^k})$1

where

$GL_c(F_{q^k})$2

is the exotic matrix Kloosterman sum. When $GL_c(F_{q^k})$3, one has $GL_c(F_{q^k})$4, $GL_c(F_{q^k})$5, and

$GL_c(F_{q^k})$6

The construction extends from a field $GL_c(F_{q^k})$7 to an étale $GL_c(F_{q^k})$8-algebra

$GL_c(F_{q^k})$9

with a composite character $GL_c(F_q)$0. In that setting the exotic matrix Kloosterman sum is defined by convolution,

$GL_c(F_q)$1

The normalizations are fixed so that classical exotic Gauss sums satisfy the Hasse–Davenport relations, exotic Kloosterman sums admit normalized forms $GL_c(F_q)$2, and the non-Abelian exotic Gauss sum carries the prefactor $GL_c(F_q)$3. These normalizations are structural: they are the form in which the reduction theorems, Hall–Littlewood formulas, and Bessel–Speh identities are stated (Zelingher, 8 Jul 2025).

2. Reduction to abelian exotic Gauss sums

A central theorem is that non-Abelian exotic Gauss sums reduce to products of abelian exotic Gauss sums. Let $GL_c(F_q)$4 be an irreducible representation of $GL_c(F_q)$5 with cuspidal support $GL_c(F_q)$6, where each $GL_c(F_q)$7 is an irreducible cuspidal representation of $GL_c(F_q)$8 corresponding to the Frobenius orbit of a regular character $GL_c(F_q)$9. Then

$F_q$0

where the abelian exotic Gauss sum

$F_q$1

factors as

$F_q$2

(Zelingher, 8 Jul 2025).

The proof uses two earlier ingredients. The first is Kondo’s explicit formula for twisted non-Abelian Gauss sums on $F_q$3: $F_q$4 The second is the Shintani lift $F_q$5 of $F_q$6 to $F_q$7, characterized by the intertwining identity

$F_q$8

where $F_q$9 is uniquely determined by $GL_c$0. This transports Kondo’s computation to the exotic setting.

For composite characters $GL_c$1 on $GL_c$2, multiplicativity becomes exact: $GL_c$3 hence

$GL_c$4

The same formalism yields a Hasse–Davenport reduction on the non-Abelian side. If $GL_c$5 and $GL_c$6, then

$GL_c$7

Translating this through the operator identity defining $GL_c$8 produces the corresponding Hasse–Davenport identity for exotic matrix Kloosterman sums: $GL_c$9 This reduction theory shows that the matrix sum is non-Abelian at the level of definition and representation theory, but abelian after factorization into regular-character data. A plausible implication is that many computational questions about $F=F_q$0 can be pushed onto explicit Gauss sums and Hasse–Davenport identities rather than handled directly on $F=F_q$1 (Zelingher, 8 Jul 2025).

3. Hall–Littlewood realization and the exotic Kloosterman sheaf

The class function $F=F_q$2 admits a geometric and symmetric-function description. Suppose $F=F_q$3 is conjugate to

$F=F_q$4

where $F=F_q$5 has degree $F=F_q$6, $F=F_q$7, $F=F_q$8, and the Frobenius orbits $F=F_q$9 are distinct. Then

$\psi:F\to \mathbf{C}$0

where $\psi:F\to \mathbf{C}$1 is the modified Hall–Littlewood polynomial and $\psi:F\to \mathbf{C}$2 are the roots of the normalized $\psi:F\to \mathbf{C}$3-function attached to Katz’s exotic Kloosterman sheaf at $\psi:F\to \mathbf{C}$4. Equivalently,

$\psi:F\to \mathbf{C}$5

with normalized roots $\psi:F\to \mathbf{C}$6 (Zelingher, 8 Jul 2025).

The sheaf-theoretic input is precise. For $\psi:F\to \mathbf{C}$7, the exotic Kloosterman local system

$\psi:F\to \mathbf{C}$8

is a rank-$\psi:F\to \mathbf{C}$9 pure local system on $k\ge 1$0 of weight $k\ge 1$1. Its normalized $k\ge 1$2-function has degree $k\ge 1$3,

$k\ge 1$4

and the normalized roots satisfy $k\ge 1$5. Moreover,

$k\ge 1$6

In the regular case $k\ge 1$7, the Hall–Littlewood polynomial simplifies to the complete homogeneous symmetric polynomial $k\ge 1$8, and

$k\ge 1$9

Thus regular classes are governed by Frobenius traces on symmetric powers of the exotic Kloosterman local system.

Macdonald’s characteristic maps organize these formulas globally. The class function $F_q$00 has an Euler-product expansion under the characteristic map on the conjugacy-class side,

$F_q$01

and under the dual characteristic map on the character side,

$F_q$02

This identifies the global exotic matrix Kloosterman function with an Euler product whose coefficients are encoded simultaneously by Hall–Littlewood combinatorics and abelian exotic Gauss sums. The representation-theoretic and sheaf-theoretic descriptions are therefore not parallel formalisms but two presentations of the same class function (Zelingher, 8 Jul 2025).

4. Bessel–Speh functions and Ginzburg–Kaplan gamma factors

A second structural axis is the relation with special values of Bessel functions attached to Speh representations. Let $F_q$03 be an irreducible generic representation of $F_q$04. For $F_q$05, its Speh representation $F_q$06 is an irreducible representation of $F_q$07 whose parameter is supported on the cuspidal components of $F_q$08 with partition $F_q$09 per component. Carmon proved that $F_q$10 admits a unique, up to scalars, $F_q$11-vector, where $F_q$12 is the unipotent radical of the block-upper parabolic of type $F_q$13. If $F_q$14 is such a vector, the Bessel–Speh function is

$F_q$15

Its special values are

$F_q$16

(Zelingher, 8 Jul 2025).

The finite-field Ginzburg–Kaplan gamma operator is

$F_q$17

where $F_q$18. By Schur’s lemma,

$F_q$19

and

$F_q$20

The link with exotic matrix Kloosterman sums is explicit. If $F_q$21 has cuspidal support $F_q$22, where each $F_q$23 corresponds to a regular character $F_q$24, and $F_q$25, then for all $F_q$26,

$F_q$27

equivalently,

$F_q$28

This identity generalizes the Curtis–Shinoda relation from the case $F_q$29 and certain principal series to Speh representations and exotic matrix Kloosterman sums.

The compatibility with $F_q$30-factors is equally concrete. For cuspidal $F_q$31 arising from regular characters $F_q$32,

$F_q$33

and multiplicativity across cuspidal supports matches the multiplicativity of the non-Abelian exotic Gauss sums. This places the Bessel–Speh identity inside a larger finite-field gamma-factor formalism. A plausible interpretation is that the exotic matrix Kloosterman function serves as a finite-field model for distinguished matrix coefficients of Speh representations, with the Ginzburg–Kaplan operator furnishing the bridge between exponential sums and tensor-product local constants (Zelingher, 8 Jul 2025).

5. Identities, multiplicativity, bounds, and examples

The theory yields a collection of exact identities. For block-diagonal matrices, let $F_q$34 and $F_q$35. Then

$F_q$36

or, for normalized sums,

$F_q$37

If $F_q$38 and $F_q$39 have no common eigenvalues in $F_q$40, then the averaging disappears and

$F_q$41

These formulas extend earlier multiplicativity statements for classical matrix Kloosterman sums (Zelingher, 8 Jul 2025).

The generating-series identity for Bessel–Speh values gives a finite-field analogue of a functional equation. For $F_q$42 and $F_q$43 irreducible generic of $F_q$44,

$F_q$45

Using the Bessel–Speh/Kloosterman identity, this becomes

$F_q$46

The Hall–Littlewood formula also gives absolute-value bounds. Since the normalized roots $F_q$47 lie on the unit circle, one obtains

$F_q$48

In the regular case $F_q$49,

$F_q$50

These are combinatorial bounds deduced from Hall–Littlewood evaluations rather than from direct cancellation estimates.

Several limiting cases recover known objects. When $F_q$51, one has

$F_q$52

which is exactly Kondo’s twisted non-Abelian Gauss sum. When $F_q$53,

$F_q$54

so the matrix theory reduces to Katz’s classical exotic Kloosterman sum on $F_q$55. For regular conjugacy classes

$F_q$56

one has $F_q$57, hence

$F_q$58

Finally, for $F_q$59 a generic principal series, the Bessel–Speh identity recovers

$F_q$60

the twisted matrix Kloosterman identity of Carmon–Zelingher 2025, now extended to general $F_q$61 with composite $F_q$62 over extensions (Zelingher, 8 Jul 2025).

6. Relation to earlier and parallel theories

The immediate antecedent is Katz’s exotic Kloosterman theory. Katz’s exotic Kloosterman sums and sheaves are sums over extensions with norm and trace conditions; the matrix theory introduces genuine matrix analogues that sum over $F_q$63 and are organized by Shintani’s norm map into conjugacy classes of $F_q$64. The non-Abelian feature is therefore not merely higher rank but the passage from scalar norm constraints to conjugacy-class data (Zelingher, 8 Jul 2025).

Two older matrix-Gauss-sum theories provide important background. "Gauss sums over some matrix groups" evaluates Gauss sums on $F_q$65 and $F_q$66 in terms of classical Gauss sums and $F_q$67-dimensional Kloosterman sums by averaging over Borel subgroups (Li et al., 2011). "Gauss sums of some matrix groups over $F_q$68" gives analogous formulas over $F_q$69, expressing $F_q$70-sums through classical Gauss sums and $F_q$71-sums through hyper-Kloosterman sums (Hu et al., 2018). These works are non-abelian in the matrix-group sense, but they are not exotic in the Deligne–Katz sense of sums over étale algebras.

A nearby extension in a different direction is "Exotic and inverted Kloosterman sums over semisimple algebras," which introduces exotic Kloosterman sums and exotic inverted Kloosterman sums attached to non-commutative finite-dimensional semisimple algebras over $F_q$72, proves reduction formulae to sums over commutative étale algebras, and obtains square-root estimates; in the inverted case an explicit correction term may appear (Wan et al., 4 Jun 2026). This setting is parallel rather than identical: the reduction passes through reduced trace and reduced norm on semisimple algebras, whereas the matrix theory of $F_q$73 passes through Shintani twisted conjugacy and representation theory.

On the representation-theoretic side, "Finite period vectors and Gauss sums" studies finite gamma factors for cuspidal representations of general linear groups over finite fields and proves product formulae in terms of abelian Gauss sums for Rankin–Selberg, Asai, exterior-square, and Bump–Friedberg factors (Jo, 2023). The formal resemblance to the reduction of $F_q$74 is direct: higher-rank gamma data are encoded by products of abelian $F_q$75-factors attached to regular characters. "On Jacobi sums arising from the classical doubling method" defines non-Abelian Jacobi sums attached to irreducible representations of general linear or classical groups over finite fields and expresses the $F_q$76 case in terms of Kondo’s non-Abelian Gauss sums; for classical groups, in the generic case, the sums are computed by Gauss sums attached to Deligne–Lusztig torus data and are constant on geometric Lusztig series (Yost-Wolff et al., 6 Dec 2025). Together with the Bessel–Speh relation, these works indicate that finite-field special-value problems for distinguished vectors, doubling kernels, and gamma operators repeatedly collapse to structured products of Gauss sums.

Within this landscape, non-Abelian exotic Gauss sums occupy a specific position. They combine Shintani descent, Kondo’s computation, Macdonald characteristic maps, Katz’s exotic sheaves, and Ginzburg–Kaplan gamma factors into one formalism. The result is a theory in which matrix exponential sums, Hall–Littlewood polynomials, and special values of Bessel–Speh functions are not separate phenomena but different realizations of the same finite-field objects (Zelingher, 8 Jul 2025).