---
title: Exotic Matrix Kloosterman Sums
url: https://www.emergentmind.com/topics/exotic-matrix-kloosterman-sums
type: topic
---

# Exotic Matrix Kloosterman Sums

Searching arXiv for recent and foundational papers on exotic matrix Kloosterman sums and closely related matrix Kloosterman constructions.
Exotic matrix Kloosterman sums are structured exponential sums in which classical Kloosterman-type oscillation is organized by matrix, group-theoretic, or algebraic data rather than by a single scalar variable. In the recent literature this phrase encompasses several related constructions: cyclotomic matrices whose entries are finite-field Kloosterman sums indexed by linear or quadratic forms [2606.01759]; matrix Kloosterman sums on $\mathrm{GL}_n(\mathbb{F}_q)$ defined by summation over invertible matrices [2109.00762]; higher-rank Bruhat-cell Kloosterman sums attached to Weyl elements in $\mathrm{SL}_3$ and $\mathrm{GL}_n$ [1109.4661], [2001.01936], [2208.01295]; non-abelian exotic matrix Kloosterman sums defined via Shintani norm maps and étale algebras [2507.06394]; and exotic or inverted Kloosterman sums over semisimple algebras, including matrix algebras, reduced to commutative étale settings [2606.05771]. Across these settings, the common theme is that additive and multiplicative structures are encoded in matrices, conjugacy classes, Weyl-group cells, or semisimple algebra decompositions, producing determinant formulas, rank phenomena, factorization identities, square-root bounds, and links to representation theory, $\ell$-adic cohomology, and automorphic analysis.

## 1. Scalar prototypes and the passage to matrix frameworks

The basic finite-field Kloosterman sum over $\mathbb{F}_p$ is
$$
K_p(a,b)=\sum_{x\in\mathbb{F}_p^\times} e^{\frac{2\pi i}{p}(ax+b x^{-1})}
=\sum_{x\in\mathbb{F}_p^\times}\zeta_p^{a x+b x^{-1}},
$$
with $\zeta_p=e^{2\pi i/p}$ and additive character $\psi(x)=e^{2\pi i x/p}$ [2606.01759]. For $a\neq 0$, one has the invariance $K_p(a,b)=K_p(1,ab)$, more generally $K_p(a,b)=K_p(\lambda a,b/\lambda)$ for $\lambda\in\mathbb{F}_p^\times$, and the Weil-type bound $|K_p(a,b)|\le 2\sqrt p$ when $ab\neq 0$ [2606.01759]. The same source abbreviates $K_p(1,b)$ to $K_p(b)$ and emphasizes the classical analogy with the modified Bessel function $K_0$.

One route from scalar to matrix behavior is to build matrices whose entries are scalar Kloosterman sums indexed by structured forms. Two principal families are
$$
A_1(p)=\bigl[K_p(i+j)\bigr]_{1\le i,j\le p-1},\qquad
A_2(p)=\bigl[K_p(i^2+j^2)\bigr]_{1\le i,j\le (p-1)/2},
$$
whose entries lie in the cyclotomic field $\mathbb{Q}(\zeta_p)$ [2606.01759]. Another route is to let the summation variable itself be a matrix. For $a\in M_n(\mathbb{F}_{q^m})$,
$$
K_n(a,\mathbb{F}_{q^m})=\sum_{x\in \mathrm{GL}_n(\mathbb{F}_{q^m})}\psi_m(a x+x^{-1}),
$$
and for $a,b\in M_n(\mathbb{F}_q)$,
$$
K_n(a,b)=\sum_{x\in \mathrm{GL}_n(\mathbb{F}_q)}\psi(a x+b x^{-1}),
$$
with $\psi=\phi\circ\mathrm{tr}$ and $\psi_m=\phi_m\circ\mathrm{tr}$ [2109.00762]. In this setting there is symmetry $K_n(a,b)=K_n(b,a)$ and $G\times G$-conjugation invariance
$$
K_n(gah^{-1},hbg^{-1})=K_n(a,b)
$$
for $g,h\in \mathrm{GL}_n(\mathbb{F}_q)$ [2109.00762].

A third route arises in automorphic and Bruhat-theoretic settings. For Weyl elements in $\mathrm{SL}_3(\mathbb{Z})$ and $\mathrm{GL}_{n+1}$, Kloosterman sums emerge from double-coset constructions and become explicit exponential sums with several moduli and multiple congruence constraints [1109.4661], [2208.01295]. In these higher-rank contexts, the term “exotic” often refers to sums associated with non-long or non-Voronoi Weyl elements, or to long-element sums whose parameterization is markedly non-abelian [2208.01295].

A fourth route, developed in the recent non-abelian theory, defines exotic matrix Kloosterman sums as class functions on $\mathrm{GL}_c(\mathbb{F}_q)$ built from Shintani norm classes and multiplicative characters on étale algebras [2507.06394]. This perspective generalizes both Katz’s exotic Kloosterman sums and twisted matrix Kloosterman sums [2507.06394].

## 2. Cyclotomic Kloosterman matrices over finite fields

For odd primes $p$, the paper on cyclotomic matrices related to Kloosterman sums studies the matrices
$$
A_1(p)=\bigl[K_p(i+j)\bigr]_{1\le i,j\le p-1},\qquad
A_2(p)=\bigl[K_p(i^2+j^2)\bigr]_{1\le i,j\le (p-1)/2}
$$
through explicit cyclotomic factorization and quadratic Gauss sum identities [2606.01759].

For $A_1(p)$, let $M_p$ be the $(p-1)\times(p-1)$ matrix $M_p(i,j)=\zeta_p^{ij}$ and let
$$
D_p=\mathrm{diag}(\zeta_p^{1/1},\zeta_p^{1/2},\dots,\zeta_p^{1/(p-1)}).
$$
Then
$$
M_p^H M_p = p I_{p-1}-J_{p-1},\qquad
\det(M_p^H M_p)=p^{p-2},
$$
and there is a key decomposition
$$
\bigl[K_p(i-j)\bigr]_{1\le i,j\le p-1}=M_p D_p M_p^H,
$$
with $\det(D_p)=1$ [2606.01759]. Passing from $[K_p(i-j)]$ to $A_1(p)=[K_p(i+j)]$ is accomplished by the column permutation $j\mapsto -j$, whose sign is
$$
\mathrm{sign}(\tau_p)=(-1)^{(p-1)/2}.
$$
Hence
$$
\det A_1(p)=(-1)^{(p-1)/2}p^{p-2},
$$
so $A_1(p)$ is nonsingular for all odd primes $p$ [2606.01759].

For $A_2(p)$, with $p=2n+1$, the argument introduces the symmetric cyclotomic matrix
$$
N_p(i,j)=\zeta_p^{2ij}+\zeta_p^{-2ij},\qquad 1\le i,j\le n,
$$
satisfying
$$
N_pN_p^T=pI_n-2J_n,\qquad \det(N_p)^2=p^{n-1},
$$
so $N_p$ is invertible [2606.01759]. The transformed matrix
$$
L_p=N_pA_2(p)N_p^T
$$
has an explicit decomposition
$$
L_p = (-1)^n p\,[\delta_p(i,j)]_{1\le i,j\le n}-v_p 1_n^T-1_n v_p^T,
$$
where the last two terms have rank at most $2$, and
$$
\delta_p(i,j)=
\begin{cases}
p,& i^2+j^2\equiv 1\pmod p,\\
0,& \text{otherwise}.
\end{cases}
$$
Using the counting lemma that the number of pairs $(i,j)$ with $1\le i,j\le n$ and $i^2+j^2\equiv 1\pmod p$ equals $(p-4-(-1)^n)/4$, one obtains
$$
\mathrm{rank}(A_2(p))\le \frac{p+4-(-1)^n}{4}.
$$
This is strictly less than $n=(p-1)/2$ when $p\ge 11$, so $A_2(p)$ is singular if and only if $p\ge 11$ [2606.01759].

The threshold behavior is explicit: $\det A_2(3)=2$, $\det A_2(5)=-5$, and $\det A_2(7)=49$, whereas $A_2(11)$ and $A_2(13)$ are singular by the rank bound [2606.01759]. The paper does not provide the characteristic polynomial or exact eigenstructure of $A_1$ or $A_2$; the determinant and rank arguments are the core results [2606.01759].

These examples are “exotic” in a restricted but concrete sense: they are not sums over matrices, but matrices built from Kloosterman sums indexed by additive and quadratic forms. Their structure is controlled by discrete Fourier-type matrices, Gauss sums, and sparse incidence conditions on the finite-field circle $i^2+j^2\equiv 1\pmod p$ [2606.01759].

## 3. Matrix Kloosterman sums on $\mathrm{GL}_n$ and cohomological purity

A direct matrix generalization replaces the scalar summation variable by $x\in\mathrm{GL}_n(\mathbb{F}_{q^m})$ and studies
$$
K_n(a,\mathbb{F}_{q^m})=\sum_{x\in\mathrm{GL}_n(\mathbb{F}_{q^m})}\psi_m(a x+x^{-1}),
$$
or the two-parameter variant
$$
K_n(a,b)=\sum_{x\in\mathrm{GL}_n(\mathbb{F}_q)}\psi(a x+b x^{-1})
$$
[2109.00762]. These sums arise in the study of expanding horospheres on $\mathrm{GL}_n$ and in effective equidistribution problems related to Marklof’s conjecture [2109.00762].

A central result is purity for regular semisimple parameters. If $a\in \mathrm{GL}_n(\mathbb{F}_q)$ is regular semisimple, then the cohomology
$$
H_c^\bullet\bigl(\mathrm{GL}_n,\; x\mapsto \mathrm{tr}(a x+x^{-1})\bigr)
$$
is pure of weight $n^2$ and concentrated in degree $n^2$ [2109.00762]. By the Grothendieck trace formula, this yields the square-root-type estimate
$$
|K_n(a)|\le 2^n q^{n^2/2}
$$
for regular semisimple $a$ [2109.00762]. The proof uses Bruhat decomposition, Künneth, Deligne’s purity for rank-$1$ Kloosterman sheaves, and a linearity lemma denoted “Hfg,” which converts affine-linear cancellations on fibers into cohomological vanishing [2109.00762].

In the split semisimple case, if $a$ is diagonal with distinct eigenvalues $\alpha_1,\dots,\alpha_n\in\mathbb{F}_q$, there is an exact product formula
$$
K_n(a)=q^{n(n-1)/2}\prod_{i=1}^n K_1(\alpha_i),
$$
where $K_1$ is the classical rank-$1$ Kloosterman sum [2109.00762]. This is a precise tensor-product factorization of the matrix sum into scalar components. The scalar case $a=\alpha I_n$ satisfies the recursion
$$
K_n(\alpha I_n)=q^{n-1}K_1(\alpha)K_{n-1}(\alpha I_{n-1})
+q^{2n-2}(q^{n-1}-1)K_{n-2}(\alpha I_{n-2}),
$$
and also admits a closed form over involutions $w\in S_n$ [2109.00762].

Beyond purity, there is a general bound valid for all $a\in\mathrm{GL}_n(\mathbb{F}_q)$:
$$
|K_n(a)|\ll_n q^{(3n^2-\delta(n))/4},
\qquad
\delta(n)=
\begin{cases}
0,& n\ \text{even},\\
1,& n\ \text{odd},
\end{cases}
$$
which is described as optimal [2109.00762]. The proof stratifies $\mathrm{GL}_n$ into Borel Bruhat cells $C_w$; cells with $w^2\neq I$ contribute vanishing, while involutive cells are controlled by an explicit inversion statistic $N(w)$ [2109.00762].

Degenerate two-parameter sums are also bounded sharply. If $r=\mathrm{rk}(b)\ge s=\mathrm{rk}(a)$ and $m=\min(r,n-r)$, then
$$
|K_n(a,b)|\le 2 q^{n^2-rn+r^2+\binom{m}{2}},
$$
and in particular, if $(a,b)\neq (0,0)$,
$$
|K_n(a,b)|\le 2 q^{n^2-n+1},
$$
with sharpness exhibited by
$$
K_n(e_{1,n},e_{1,n})
= q^{2n-2}|\mathrm{GL}_{n-2}(\mathbb{F}_q)|
+(q-1)q^{n-1}|\mathrm{GL}_{n-1}(\mathbb{F}_q)|
\sim q^{n^2-n+1}
$$
[2109.00762].

This line of work shows that matrix Kloosterman sums are not merely higher-dimensional analogues of scalar sums. They are controlled by a mixture of Bruhat combinatorics, perverse-sheaf purity, and explicit rank-$1$ Kloosterman sheaf factors, and they furnish the analytic input needed for higher-rank horospherical equidistribution [2109.00762].

## 4. Spectral and linear-algebraic viewpoints on Kloosterman matrices

A different but related matrix theory fixes a modulus and places classical Kloosterman sums into a matrix indexed by residue classes. For a positive integer $q$, define the $q\times q$ Kloosterman matrix
$$
K_q=(S(m,n;q))_{0\le m,n\le q-1},
$$
where
$$
S(m,n;q)=\sum_{\substack{a\!\!\!\pmod q\\(a,q)=1}}
\exp\!\left(\frac{2\pi i}{q}(am+\bar a\,n)\right)
$$
[1803.02970]. In the Fourier basis $v^{(j)}=(e^{2\pi i jm/q})_{m=0}^{q-1}$, one has
$$
K_qv^{(j)}=
\begin{cases}
q\,v^{(-\bar j)},&(j,q)=1,\\
0,&(j,q)>1,
\end{cases}
$$
so the action on coprime Fourier modes is the involution $j\mapsto -\bar j$ [1803.02970].

Consequently, the nonzero spectrum of $K_q$ is supported on $\{\pm q\}$, while $0$ has multiplicity $q-\varphi(q)$ [1803.02970]. Writing
$$
F=\widetilde\varphi(q)=\#\{x\in (\mathbb{Z}/q\mathbb{Z})^\times: x^2\equiv -1\pmod q\},
$$
the multiplicities are
$$
\mathrm{mult}(+q)=\frac{\varphi(q)+F}{2},\qquad
\mathrm{mult}(-q)=\frac{\varphi(q)-F}{2},\qquad
\mathrm{mult}(0)=q-\varphi(q)
$$
[1803.02970]. Equivalently,
$$
\det(\lambda I_q-K_q)
=
\lambda^{q-\varphi(q)}
(\lambda-q)^{(\varphi(q)+\widetilde\varphi(q))/2}
(\lambda+q)^{(\varphi(q)-\widetilde\varphi(q))/2}.
$$
The matrix satisfies
$$
K_q^2=q C_q,
$$
where $C_q$ is the Ramanujan matrix $C_q=(c_q(m-n))_{m,n}$ [1803.02970].

This spectral picture is “exotic” because pointwise Weil-type bounds on $S(m,n;q)$ do not govern the operator norm. Instead,
$$
\|K_q\|_{\mathrm{op}}=q,\qquad \mathrm{rank}(K_q)=\varphi(q),
$$
and the structure is dictated by a permutation on Fourier modes, producing a large kernel together with a two-point nonzero spectrum [1803.02970]. The same phenomenon persists for matrices obtained by summing Kloosterman sums over moduli up to $Q$; the spectrum is again supported on $\{-x,0,x\}$ with $x=\mathrm{lcm}(1,\dots,Q)$ [1803.02970].

A related but distinct finite-field matrix-group setting appears in Gauss sums over $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{SL}_n(\mathbb{F}_q)$. For $U\in M_n(\mathbb{F}_q)$, the sum
$$
G_{\mathrm{SL}_n(\mathbb{F}_q)}(U,\psi)=\sum_{X\in \mathrm{SL}_n(\mathbb{F}_q)}\psi(\mathrm{tr}(U^tX))
$$
reduces, when $\mathrm{rank}(U)=n$, to
$$
G_{\mathrm{SL}_n(\mathbb{F}_q)}(U,\psi)
=
- q^{n(n-1)/2} K_n(1,\det U),
$$
where
$$
K_n(1,y)=\sum_{x_1\cdots x_n=y}\psi(x_1+\cdots+x_n)
$$
is the $n$-dimensional hyper-Kloosterman sum [1105.4513]. This is not the same object as $K_n(a)$ on $\mathrm{GL}_n(\mathbb{F}_q)$ in the matrix-summation sense, but it exemplifies a matrix-group exponential sum collapsing to a hyper-Kloosterman sum through Borel averaging [1105.4513].

## 5. Higher-rank Bruhat-cell and Weyl-element Kloosterman sums

In rank $2$ and higher, Kloosterman sums attached to Weyl elements become fundamentally multi-parameter. For $\mathrm{SL}_3(\mathbb{Z})$, the long-element Kloosterman sum can be written explicitly in Plücker coordinates as a sum over quadruples $(B_1,C_1;B_2,C_2)$ subject to coprimality and congruence conditions, with phase depending on Bézout coefficients $Y_i,Z_i$ [1109.4661]. These sums have two moduli $(A_1,A_2)$ and embody the non-abelian geometry of the long Weyl cell.

A weighted average of long-element $\mathrm{SL}_3(\mathbb{Z})$ Kloosterman sums satisfies the bound
$$
\sum_{v\in\{\pm1\}^2}\sum_{c_1,c_2\ge 1}
\frac{S_{w_\ell}(\psi_m,\psi_{vn},(c_1,c_2))}{c_1c_2}\,
f\!\left(
X\frac{\pi c_2m_1n_2}{c_1^2},
Y\frac{\pi c_1m_2n_1}{c_2^2}
\right)
\ll_{f,m,n,\varepsilon}
(XY)^\varepsilon\big((XY)^{5/14}+X^{1/2}+Y^{1/2}\big),
$$
for compactly supported $f$ that is eight-times differentiable in each variable, in the regime
$$
\frac12<\frac{\log c_1}{\log c_2}<2
$$
[1109.4661]. The mechanism is cancellation from variation in argument, accessed via Li’s Kuznetsov formula on $\mathrm{SL}_3(\mathbb{R})$, Mellin–Barnes representations of the kernel functions, and a partial inversion formula [1109.4661].

A more arithmetic decomposition of the $\mathrm{SL}_3$ long-word Kloosterman sum is obtained by stratifying the big cell according to reduced words and gcd data [2001.01936]. Writing the coarse long-word sum as a sum over fine strata indexed by $(d_1,d_2,f)$,
$$
S_{w_0}(m,n;(c_1,c_2))
=
\sum_{f\mid \gcd(c_1,c_2)}
S_{w_0}\!\left(m,n;\frac{c_1}{f},\frac{c_2}{f},f\right),
$$
the fine sum vanishes unless
$$
(m_2d_2,f)=(n_2d_1,f),
$$
and when this condition holds,
$$
S_{w_0}(m,n;d_1,d_2,f)
=
f\!\!\!\sum_{\substack{x,y\bmod f\\xy\equiv1\!\!\!\!\!\pmod f\\m_2d_2+n_2d_1y\equiv 0\!\!\!\!\!\pmod f}}
S\!\left(n_1,\frac{m_2d_2+n_2d_1y}{f};d_1\right)
S\!\left(m_1,\frac{m_2d_2x+n_2d_1}{f};d_2\right)
$$
[2001.01936]. Thus the fine $\mathrm{SL}_3$ long-word sum is an explicit finite sum of products of two classical $\mathrm{GL}_2$ Kloosterman sums [2001.01936].

This yields sharper bounds than earlier long-word estimates. For the coarse sum,
$$
|S_{w_0}(m,n;(c_1,c_2))|
\le
\sqrt{c_1c_2}\,(c_1,c_2)^{1/2}\tau((c_1,c_2))\tau(c_1)\tau(c_2)\cdot \min\{A,B\},
$$
with
$$
A=(m_2n_1,c_1)^{1/2}(n_2m_1,c_2)^{1/2},\qquad
B=(m_2n_1,c_2)^{1/2}(n_2m_1,c_1)^{1/2}
$$
[2001.01936].

For $\mathrm{GL}(n+1)$, local Kloosterman sums associated with Weyl elements admit explicit exponential-sum representations. The long Weyl element $w_1$ and an order-$2$ element $w^\ast$ are treated in detail, with the exact decompositions
$$
\mathrm{Kl}_p(\psi,\psi',C^\ast w_1)=\sum_{m\in M_{w_1}(r)} \mathrm{Kl}_p(m,\psi,\psi',w_1),
$$
and
$$
\mathrm{Kl}_p(\psi,\psi',C^\ast w^\ast)=\sum_{m\in M_{w^\ast}(r)} \mathrm{Kl}_p(m,\psi,\psi',w^\ast)
$$
[2208.01295]. These partial sums are nested exponential sums with explicit $p$-adic moduli and coprimality conditions.

The main conclusion is power saving over the Dąbrowski–Reeder trivial bound:
$$
\mathrm{Kl}_p(\psi,\psi',C^\ast w_1)\ll \Big(\prod_{k=1}^n p^{r_k}\Big)^{1-\delta},
\qquad
\mathrm{Kl}_p(\psi,\psi',C^\ast w^\ast)\ll \Big(\prod_{k=1}^n p^{r_k}\Big)^{1-\delta},
$$
with $\delta\gg 1/n^2$ for $w_1$ and $\delta\gg 1/n$ for $w^\ast$ [2208.01295]. For $C=(p,\dots,p)$ and $p\nmid v_jv'_j$ for $1<j<n$,
$$
\mathrm{Kl}_p(\psi,\psi',C^\ast w^\ast)=p^{n-1}+p^{n-2},
$$
showing that the order-$2$ case can be large and that the power-saving exponent must depend on the Weyl element [2208.01295]. The same paper treats all nontrivial Weyl classes in $\mathrm{GL}(4)$ and applies the resulting bounds to go beyond Sarnak’s density conjecture for the principal congruence subgroup of prime level [2208.01295].

## 6. Non-abelian exotic matrix Kloosterman sums and semisimple-algebra variants

A recent representation-theoretic formalism defines exotic matrix Kloosterman sums as class functions on $\mathrm{GL}_c(\mathbb{F}_q)$ built from characters on finite extensions and the Shintani norm map [2507.06394]. For a character $\chi:\mathbb{F}_{q^k}\to \mathbb{C}$, the basic class function is
$$
\mathrm{Kl}(\chi,\psi,h)=
\sum_{x\in \mathrm{GL}_c(\mathbb{F}_{q^k}):\, h\in \mathrm{Cl}(N_{k/1}(x))}
\frac{1}{|\mathrm{Cl}(N_{k/1}(x))|}
\chi(\det x)\psi_k(\mathrm{tr}\,x),
$$
and for an étale algebra parameter $\lambda=(k_1,\dots,k_s)$ with $\alpha=\alpha_1\times\cdots\times \alpha_s$, the exotic matrix Kloosterman sum is defined by convolution
$$
\mathrm{Kl}(\alpha,\psi,h)
=
\sum_{h_1\cdots h_s=h}\prod_{i=1}^s \mathrm{Kl}(\alpha_i,\psi,h_i)
$$
[2507.06394].

The associated non-abelian exotic Gauss transform is
$$
G_\pi^\chi(\psi)=q^{-kc^2/2}\sum_{h\in \mathrm{GL}_c(\mathbb{F}_q)}\mathrm{Kl}(\chi,\psi,h)\,\pi(h),
$$
for irreducible $\pi$ of $\mathrm{GL}_c(\mathbb{F}_q)$ [2507.06394]. By Shintani lift, this equals Kondo’s non-abelian Gauss sum on the Shintani lift $\Pi$ of $\pi$:
$$
G_\pi^\chi(\psi)=G_\Pi^\chi(\psi_k)
$$
[2507.06394]. If the cuspidal support of $\pi$ is $\{\pi_1,\dots,\pi_t\}$ with corresponding Frobenius orbits $[\beta_j]$, then
$$
G_\pi^\chi(\psi)=q^{-kc/2}(-1)^c\prod_{j=1}^t G_{c_j,k}(\beta_j,\chi;\psi),
$$
and for composite $\alpha$ one has
$$
G_\pi^\alpha(\psi)=q^{-kc/2}(-1)^{cs}\prod_{i=1}^s\prod_{j=1}^t G_{c_j,k_i}(\beta_j,\alpha_i;\psi)
$$
[2507.06394].

The central structural theorem expresses the class function $\mathrm{Kl}(\alpha,\psi,h)$ in terms of modified Hall–Littlewood polynomials evaluated at Frobenius roots of Katz’s exotic Kloosterman sheaf. If
$$
h\sim \mathrm{diag}\bigl(J_{\mu_1}(h_{\xi_1}),\dots,J_{\mu_r}(h_{\xi_r})\bigr),
$$
with distinct Frobenius orbits $[\xi_i]$, then
$$
\mathrm{Kl}(\alpha,\psi,h)
=
(-1)^{(k-1)c}q^{(k-1)\binom c2}
\prod_{i=1}^r
\widetilde H_{\mu_i}(\omega_{1,[\xi_i]},\dots,\omega_{k,[\xi_i]};q^{a_i}),
$$
equivalently in normalized roots,
$$
\mathrm{Kl}^\ast(\alpha,\psi,h)
=
\prod_{i=1}^r
\widetilde H_{\mu_i}\bigl(((-1)^{k-1}\omega_{1,[\xi_i]}^\ast,\dots,(-1)^{k-1}\omega_{k,[\xi_i]}^\ast);q^{a_i}\bigr)
$$
[2507.06394]. For regular elements this becomes a product of traces on symmetric powers of the exotic Kloosterman sheaf [2507.06394].

The same theory relates exotic matrix Kloosterman sums to special values of Bessel functions attached to Speh representations. If $\tau$ is generic with cuspidal support corresponding to $\alpha=\times \alpha_j$, then
$$
\mathfrak{B}_\tau(h)
=
(-1)^{(k+s)c}q^{-(k-1)c^2}\,
\mathrm{Kl}(\alpha^{-1},\psi,(-1)^{k-1}h^{-1}),
$$
or in normalized form,
$$
\mathfrak{B}_\tau^\ast(h)=(-1)^{(k+s)c}\,\mathrm{Kl}(\alpha^{-1},\psi,(-1)^{k-1}h^{-1})
$$
[2507.06394]. This yields multiplicativity and generating-series identities, and a bound
$$
|\mathrm{Kl}^\ast(\alpha,\psi,h)|
\le
\prod_{i=1}^r \#\{\text{weak flags in }\overline{\mathbb{F}}_{q^{a_i}}^{\,b_i}\text{ of length }k\text{ stable under }J_{\mu_i}(1)\},
$$
with the simpler estimate
$$
|\mathrm{Kl}^\ast(\alpha,\psi,h)|\le \prod_{i=1}^r \binom{b_i+k-1}{b_i}
$$
for regular $h$ [2507.06394].

A complementary 2026 development defines exotic Kloosterman and exotic inverted Kloosterman sums on finite-dimensional semisimple $\mathbb{F}_q$-algebras $M$, using reduced norm and reduced trace:
$$
\mathrm{Kl}_M(a;\chi)=\sum_{\substack{x\in M^\times\\ \mathrm{Nrd}(x)=a}}\chi(x)\psi(\mathrm{Trd}(x)),
$$
$$
\mathrm{IKl}_M(a;\chi)=\sum_{\substack{x\in M^\times\\ \mathrm{Nrd}(x)=a\\ \mathrm{Trd}(x)\neq 0}}
\chi(x)\psi\!\left(\frac{1}{\mathrm{Trd}(x)}\right)
$$
[2606.05771]. When
$$
M\simeq \prod_{i=1}^k M_{n_i}(\mathbb{F}_{q^{d_i}}),
$$
with determinant-type character $\chi$, there is an exact reduction to the commutative étale algebra
$$
B'=\prod_{i=1}^k \mathbb{F}_{q^{d_i}}^{\,n_i}
$$
and transferred character $\eta$:
$$
\mathrm{Kl}_M(a;\chi)=q^N\,\mathrm{Kl}_{B'}(a;\eta),
\qquad
N=\sum_{i=1}^k d_i\binom{n_i}{2}
$$
[2606.05771]. For inverted sums, either
$$
\mathrm{IKl}_M(a;\chi)=q^N\,\mathrm{IKl}_{B'}(a;\eta),
$$
or, if $\eta=\rho\circ \nu$ for a character $\rho$ of $\mathbb{F}_q^\times$, there is an explicit correction term:
$$
\mathrm{IKl}_M(a;\chi)
=
q^N\,\mathrm{IKl}_{B'}(a;\eta)
-\rho(a)\,\frac{\#M^\times-\#\mathcal{B}_M}{q(q-1)}
$$
[2606.05771].

For $M=M_n(\mathbb{F}_q)$ and $\chi(A)=\omega(\det A)$,
$$
\mathrm{Kl}_{M_n(\mathbb{F}_q)}(a;\chi)
=
q^{\binom n2}
\sum_{\substack{x_1,\dots,x_n\in \mathbb{F}_q^\times\\ x_1\cdots x_n=a}}
\omega(x_1)\cdots\omega(x_n)\psi(x_1+\cdots+x_n),
$$
so the matrix-algebra sum reduces to a twisted hyper-Kloosterman sum [2606.05771]. In particular, for $\omega=1$,
$$
\mathrm{Kl}_{M_n(\mathbb{F}_q)}(a;1)=q^{\binom n2}\,\mathrm{Kl}_n(a),
$$
recovering Kim’s formula [2606.05771].

The reduction yields square-root bounds. If $\dim_{\mathbb{F}_q}M=\sum_i d_i n_i^2$ and $\dim_{\mathbb{F}_q}B'=\sum_i d_i n_i$, then
$$
|\mathrm{Kl}_M(a;\chi)|
\le
(\dim_{\mathbb{F}_q}B')\,q^{(\dim_{\mathbb{F}_q}M-1)/2}
$$
[2606.05771]. For $M=M_n(\mathbb{F}_q)$ this becomes
$$
|\mathrm{Kl}_{M_n(\mathbb{F}_q)}(a;\chi)|\le n\,q^{(n^2-1)/2}
$$
by direct substitution into the stated formula [2606.05771]. The inverted sums admit analogous bounds, with the explicit correction term when $\eta=\rho\circ\nu$ [2606.05771].

## 7. Analytic applications, misconceptions, and open directions

Several distinct applications motivate these constructions. Matrix Kloosterman sums on $\mathrm{GL}_n(\mathbb{F}_q)$ provide the power savings needed for effective equidistribution of primitive rational points on expanding horospheres, and have already been used by El-Baz–Lee–Strömbergsson in higher-rank dynamics [2109.00762]. Higher-rank Weyl-element Kloosterman sums enter Kuznetsov formulae, density problems, and moment calculations; the new bounds for long and order-$2$ Weyl elements in $\mathrm{GL}(n+1)$ are explicitly applied to exceed the density exponent predicted by Sarnak’s conjecture at prime level for $n>5$ [2208.01295]. The $\mathrm{SL}_3$ long-word theory feeds trace formulas and explicit double Dirichlet-series formulae for the triple divisor function [2001.01936]. Non-abelian amplification on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ yields savings for bilinear forms in classical Kloosterman sums at composite moduli, framing Kloosterman matrices through non-abelian Fourier coefficients and sifted representations [2511.08445].

A common misconception is that “matrix Kloosterman sum” refers to a single canonical object. The literature supports several inequivalent usages. In one usage, the summation variable is a matrix in $\mathrm{GL}_n(\mathbb{F}_q)$ [2109.00762]. In another, one studies matrices whose entries are scalar Kloosterman sums [2606.01759], [1803.02970]. In yet another, matrix-group or Weyl-group geometry organizes higher-rank Kloosterman sums attached to Bruhat cells [1109.4661], [2208.01295]. Recent “exotic” terminology adds further specificity: either sums built from Shintani norms and Katz’s exotic Kloosterman sheaves [2507.06394], or reduced-trace/reduced-norm sums on semisimple algebras [2606.05771]. These are related by theme rather than by a single universal definition.

Another plausible misconception is that “exotic” simply means “higher-dimensional.” The sources suggest a narrower implication: nonstandard indexing, non-abelian or semisimple-algebra structure, reduced-word or Shintani parametrization, or unusual spectral behavior. In the cyclotomic matrix setting, “exotic” reflects the singularity threshold and the use of quadratic forms in the indices [2606.01759]. In the Weyl-element setting, it marks Kloosterman sums beyond the classical long/Voronoi template [2208.01295]. In the non-abelian theory, it signals the passage from scalar fields to étale algebras and Frobenius-orbit data [2507.06394].

The present literature also delineates clear limitations. The cyclotomic analysis in [2606.01759] is restricted to odd primes and to the specific forms $i+j$ and $i^2+j^2$. Exact ranks and eigenstructures of $A_2(p)$ for $p\ge 11$ are not determined there. The full $\mathrm{GL}_n$ matrix Kloosterman theory gives purity in the regular semisimple case and optimal global bounds, but explicit evaluations outside special classes remain rare [2109.00762]. The fine $\mathrm{SL}_3$ reduced-word stratification does not yet yield an equally attractive formula for the further intersection refinement $\Omega_{w_0}^!$ [2001.01936]. The semisimple-algebra reduction in [2606.05771] is formulated for determinant-type characters, and the strongest explicit subgroup-type bounds in the matrix-power literature remain concentrated in prime-field settings [2110.10941].

Several open directions are directly suggested by the cited works. For cyclotomic Kloosterman matrices, determining the exact rank and eigenstructure of $A_2(p)$ for $p\ge 11$, and extending the analysis to other forms such as $ij$, $i^2-j^2$, or higher-degree polynomials, remain natural problems [2606.01759]. For matrix Kloosterman sums on $\mathrm{GL}_n$, the non-split regular semisimple evaluation conjectured in [2109.00762] suggests a deeper relationship with Hall–Littlewood polynomials, a direction already developed in the exotic framework of [2507.06394]. For higher-rank Weyl-element sums, improving the $n$-dependence of the power-saving exponent $\delta$ and extending explicit LNR factorizations to more Weyl classes remain open [2208.01295]. For non-abelian exotic sums, a plausible implication is that the Hall–Littlewood and Bessel–Speh connections provide a template for further cohomological interpretations beyond the cases currently established [2507.06394].

Taken together, these results show that exotic matrix Kloosterman sums form a broad research area at the intersection of analytic number theory, algebraic geometry, finite-group representation theory, and automorphic forms. The unifying pattern is that classical oscillation of the phase $x+x^{-1}$ persists, but the ambient structure shifts from one-dimensional multiplicative groups to cyclotomic matrices, Bruhat cells, conjugacy classes, semisimple algebras, and spectral actions. The resulting theory combines determinant identities, purity theorems, product factorizations, exact reductions, and power-saving bounds in ways that are specific to the matrix or non-abelian context [2606.01759], [2109.00762], [1803.02970], [1109.4661], [2001.01936], [2208.01295], [2507.06394], [2606.05771].

Source: https://www.emergentmind.com/topics/exotic-matrix-kloosterman-sums