A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations

1. Problem Statement

The paper investigates the structure of Bernstein blocks in the category of smooth complex representations of non-archimedean reductive groups. Specifically, it focuses on two distinct but related scenarios involving the group $SL_8(F)$ (where $F$ is a non-archimedean local field with residue field order $q$ such that $4 \mid q-1$):

1. The Twisted Case ($G_0$): A twisted Levi subgroup $G_0$ of $SL_8$ arising from quadratic and quartic field extensions, associated with a depth-zero cuspidal pair $(M_0, \sigma_0)$.
2. The Classical Case ($G$): The full group $G = SL_8(F)$ associated with a supercuspidal pair $(M, \sigma)$ constructed from the same field extensions.

The central problem is to explicitly describe the Hecke algebras ($H_0$ and $H_s$) governing these blocks, determine their simple modules, and analyze the topology of their primitive spectra (Bernstein varieties). The authors aim to demonstrate that a non-trivial 2-cocycle in the Hecke algebra leads to a specific non-orientable geometric structure—the Klein bottle—within the tempered dual.

2. Methodology

The authors employ a combination of representation theory, group cohomology, and algebraic geometry:

• Hecke Algebra Construction: They utilize the theory of types (Bushnell-Kutzko) to identify compact open subgroups $K$ and representations $\rho$ that generate the Bernstein blocks. This reduces the study of the representation category to the module category of a specific Hecke algebra.
• Twisted Group Algebras: The Hecke algebras are identified as twisted group algebras of the form $C[\Gamma, \mu]$, where $\Gamma$ is an affine Weyl group extended by a central torus, and $\mu$ is a non-trivial 2-cocycle.
• Cohomological Analysis: They compute the second cohomology group $H^2(\Gamma, \mathbb{C}^\times)$ to classify the cocycles. They explicitly construct a normalized 2-cocycle $\mu$ and analyze its effect on the commutation relations of the algebra generators.
• Clifford Theory: To find simple modules, they restrict representations to a commutative subalgebra (the Bernstein torus) and induce them up to the full algebra, analyzing the orbits of the Weyl group action on the character variety.
• Topological Analysis: They examine the maximal compact real forms of the complex algebraic varieties (Bernstein varieties) to determine their topological type (e.g., torus vs. Klein bottle).

3. Key Contributions and Results

A. The Twisted Hecke Algebra ($H_0$) for $G_0$

• Structure: The Hecke algebra $H_0$ associated with the twisted Levi $G_0$ is isomorphic to the twisted group algebra $C[\Gamma, \mu]$, where $\Gamma \cong (Z \rtimes Z/2) \times Z$.
• The Cocycle: The twist is defined by a non-trivial 2-cocycle $\mu$ (specifically $\mu((\epsilon, m), (\delta, n)) = (-1)^{\epsilon n}$). This cocycle modifies the commutation relation between the $Z/2$ generator $s$ and the $Z$ generator $Y$ from $sY = Ys$ to $sY = -Ys$.
• Simple Modules:
  • Unlike the untwisted case, all simple modules of $H_0$ are 2-dimensional.
  • There are no 1-dimensional representations because the relation $sY = -Ys$ forces any scalar representation to have $Y=0$, contradicting invertibility.
  • The modules are parametrized by $(w, z) \in (\mathbb{C}^\times)^2$ with the identification $(w, z) \sim (-w, z^{-1})$.
• The Klein Bottle: The primitive spectrum (the set of isomorphism classes of simple modules) is the complex variety $X = (\mathbb{C}^\times \times \mathbb{C}^\times) / \sim$. The maximal compact real form of this variety (where $|w|=|z|=1$) is the quotient of the torus $S^1 \times S^1$ by the free, orientation-reversing involution $\tau(w, z) = (-w, z^{-1})$.
  • Result: This quotient is homeomorphic to a Klein bottle ($K_b$).
  • Significance: The Klein bottle is a model for the unitary principal series of $G_0$ attached to $(M_0, \sigma_0)$.

B. The Classical Case ($H_s$) for $SL_8$

• Comparison: The authors compare the above with the classical case of $G = SL_8(F)$ and a cuspidal pair $(M, \sigma)$ (following Roche's work).
• Isomorphism: They prove that the Bernstein varieties for the twisted case ($s_0$) and the classical case ($s$) are isomorphic:
  $$T^{s_0}/W^{s_0} \cong T^s/W^s$$
• Re-appearance of the Klein Bottle: In the classical case, the Hecke algebra also involves a non-trivial cocycle (arising from the specific construction of the supercuspidal representation). Consequently, the tempered dual of $SL_8$ also contains a component homeomorphic to the Klein bottle.
• Geometric Footprint: The paper establishes that the Klein bottle is not an anomaly but a "geometric footprint" of the non-trivial cohomology class $[\mu] \in H^2(\Gamma, \mathbb{C}^\times)$ in the Hecke algebra.

C. Technical Details on the Cocycle

• The paper clarifies that the infinite dihedral group $D_\infty$ and the cyclic group $Z/2$ individually have trivial $H^2$ with coefficients in $\mathbb{C}^\times$. The non-triviality arises specifically from the interaction between the $Z/2$ factor and the $Z$ factor in the product group $Z/2 \times Z$.
• The cocycle $\mu$ acts as a "twist" that eliminates fixed points in the Weyl group action on the character torus. In the untwisted case, fixed points yield 1-dimensional representations and a torus with boundary (or singularities); in the twisted case, the action is free, yielding a smooth, non-orientable surface.

4. Significance

1. Geometric Realization of Cohomology: The paper provides a concrete topological manifestation of non-trivial group cohomology in the context of the Langlands program. It shows that a non-trivial 2-cocycle in a Hecke algebra directly translates to a non-orientable manifold (the Klein bottle) in the tempered dual.
2. Unification of Cases: It bridges the gap between "twisted" Levi subgroups (often studied in the context of endoscopy or twisted groups) and classical supercuspidal representations of $SL_n$, showing they share the same underlying geometric structure in their Bernstein varieties.
3. Dimensionality Shift: It highlights a fundamental shift in representation theory: the presence of a specific cocycle forces all simple modules to jump from dimension 1 to dimension 2, fundamentally altering the structure of the category.
4. Explicit Classification: The authors provide explicit generators, relations, and matrix representations for the simple modules, offering a complete classification for these specific blocks of $SL_8$.

Conclusion

Aubert and Plymen demonstrate that for specific cuspidal pairs in $SL_8$ (both twisted and classical), the associated Bernstein block is governed by a twisted Hecke algebra. The non-triviality of the twisting cocycle forces the primitive spectrum to be a complex variety whose maximal compact real form is a Klein bottle. This result identifies the Klein bottle as a natural geometric object appearing in the tempered dual of reductive groups over local fields, serving as a topological signature of non-trivial cohomology in the Hecke algebra.