Iwahori-Spherical Genuine Representations

• Iwahori-spherical genuine representations are irreducible smooth representations of central extensions of p-adic groups, characterized by nontrivial Iwahori invariants and genuine Hecke algebra actions.
• Their study leverages the genuine Iwahori–Hecke algebra defined via Iwahori–Matsumoto and Bernstein presentations, enabling explicit calculations of invariants and intertwining maps.
• Langlands parametrization, detailed Whittaker model analyses, and combinatorial techniques underpin formal degree formulas and geometric wavefront set determinations.

Iwahori-spherical genuine representations are irreducible smooth representations of central extensions of split reductive $p$-adic groups by finite cyclic groups, exhibiting nontrivial invariants under an Iwahori subgroup and equivariance under a fixed embedding of the covering kernel. Their study distinctly intertwines the structure of Brylinski–Deligne covers, the theory of genuine Hecke algebras, Langlands-type parametrization, character expansions, and the combinatorics of Whittaker models and wavefront sets.

1. Brylinski–Deligne Central Covers and Genuine Representations

Let $F$ denote a non-Archimedean local field, and $G$ a split semisimple $F$-group. An $n$-fold Brylinski–Deligne cover is specified by

$$1 \rightarrow \mu_n \rightarrow \widetilde G \rightarrow G \rightarrow 1$$

where $\mu_n$ is the group of $n$th roots of unity. Fix an embedding $\zeta: \mu_n \rightarrow \mathbb{C}^\times$. A smooth irreducible representation $(\pi, V)$ of $\widetilde G$ is called $\zeta$-genuine if $\mu_n$ acts via $\zeta$.

An Iwahori subgroup $I \subset G$ splits in $\widetilde G$ and yields the genuine Iwahori–Hecke algebra

$$\mathcal{H}_\zeta(\widetilde G, I) = C_c^\infty(I \backslash \widetilde G / I)^{\zeta\text{–genuine}}$$

There is a bijection between irreducible genuine Iwahori–spherical $\widetilde G$-modules, i.e., irreducible $\zeta$-genuine representations $(\pi, V)$ with $V^I \ne 0$, and irreducible modules over $\mathcal{H}_\zeta(\widetilde G, I)$ (Dong et al., 2023).

2. Genuine Iwahori–Hecke Algebras: Structure and Presentations

The pro-$p$ Iwahori–Hecke algebra $\mathcal{H}(\widetilde G, I^1)$, with $I^1$ the maximal pro-$p$ normal subgroup of $I$, is fundamental for investigation. This $\mathbb{C}$-algebra admits both the Iwahori–Matsumoto and Bernstein presentations, employing generators indexed by the extended affine Weyl group $W_{\text{ext}}$ and the commutative subalgebra generated by coweights (Gao et al., 2022):

• Iwahori–Matsumoto presentation: Basis elements $T_w$, quadratic and braid relations encode convolution structure, reflecting the influence of the covering via extra Gauss sum and sign factors.
• Bernstein presentation: The algebra decomposes as $$\mathcal{H}_K \otimes_{\mathbb{C}[T(F)/T(\mathcal{O})^1]} \mathbb{C}[X_*(T)]$$, with explicit cross-relations and twisted multiplication governed by the underlying cover.

This structure facilitates explicit calculations of invariants and intertwining operators in genuine representations.

3. Langlands Parametrization and the Hiraga–Ichino–Ikeda Formal Degree Formula

Square-integrable Iwahori–spherical genuine representations $(\pi, V)$ are attached to enhanced Langlands parameters $(\phi, \tau)$, constructed via an isomorphism between the genuine Hecke algebra of the cover and that of a split reductive linear group $G_{Q, n}$, leveraging the Kazhdan–Lusztig–Reeder parametrization (Dong et al., 2023).

The Hiraga–Ichino–Ikeda formal degree conjecture is verified for square-integrable genuine Iwahori–spherical representations:

$$\deg(\pi, \mu_G) = C(G, \widetilde G) \cdot \frac{|\gamma(0, \pi, \operatorname{Ad}, \psi)|}{|S_\pi|}$$

where:

• $\gamma(0, \pi, \operatorname{Ad}, \psi)$ is the local gamma factor for $\operatorname{Ad} \circ \phi$,
• $S_\pi$ is the component group of the centralizer of $\phi$ modulo center,
• $$C(G, \widetilde G) = q^{\frac{\dim G}{2}} (1-q^{-1})^r$$ normalizes measures.

The verification exploits the reduction to the linear case via isomorphism, matching Hecke generators, affine lengths, and Haar measures. The formula highlights that formal degrees for genuine representations are governed by their Langlands parameters modulo central and covering phenomena (Dong et al., 2023).

4. Whittaker Models and Dimensions for Iwahori-Spherical Genuine Representations

Whittaker dimensions are pivotal in classifying genericity of genuine representations. For an almost simple, simply connected $G$ and an “oasitic” cover $\widetilde G$, one investigates the dimension of $\mathrm{Hom}_{U^-}(\pi_o, \psi)$, with $\psi$ a nondegenerate character of conductor $\mathfrak{p}_F$. For the Steinberg-type (sign) Hecke character $\varepsilon_H$, explicit formulas yield (Dong et al., 2023):

$$\dim \mathrm{Wh}_\psi(\pi_{\varepsilon_H}) = |W|^{-1} \prod_{j=1}^r (n + m_j)$$

where $m_j$ are the Weyl group exponents; analogous polynomial formulas hold for one-dimensional characters in types B, C, $F_4$, $G_2$.

Gelfand–Graev modules and Iwahori components are instrumental in these calculations, realizing Whittaker dimensions as traces of intertwining operators and explicit Weyl–group combinatorics (Gao et al., 2022, Dong et al., 2023).

5. Geometric Wavefront Sets and Nilpotent Support

For Iwahori–spherical genuine representations with positive real Satake parameters, the geometric wavefront set, defined via the Harish–Chandra expansion, reflects the maximal nilpotent orbits present in the character (Gao et al., 22 Jan 2026). The upper bound for the geometric wavefront is given by the covering Barbasch–Vogan duality:

$$\mathrm{WF}^{\rm geo}(\mathrm{AZ}(\pi)) \subseteq \{ d_{BV, G}^{(n)}(\mathcal O(\phi_\pi)) \}$$

with equality in type A groups and for some exceptional covers. For regular principal series constituents, one obtains:

$$\mathrm{WF}^{\rm geo}(\pi_S) = d_{BV, G}^{(n)}( \mathsf{Sat}_{M_S^\vee}^{\vee}(\mathcal{O}_{\mathrm{reg}}) ) = d_{BV, G}^{(n)}( \mathcal{O}(\phi_{\mathrm{AZ}(\pi_S)}) )$$

The geometric wavefront set thus encodes the interplay between Hecke algebraic reduction, Langlands parameters, and the covering structure. Leading coefficients in the character expansion relate to the dimensions of degenerate Whittaker models, articulated via geometric Springer–theoretic induction (Gao et al., 22 Jan 2026).

6. Combinatorics: Colored Lusztig Data, Gelfand–Tsetlin Patterns, and Lattice Models

The explicit evaluation of Iwahori Whittaker functions for unramified genuine principal series leverages colored Lusztig data—a refinement of classical Lusztig data by “coloring” choices associated to binary conjugation decisions in explicit Iwahori decompositions of maximal unipotent subgroups (Naprienko, 2021):

• Colored Lusztig datum: A pair $(m, \sigma)$, $m\in\Lu(\lambda+\rho)$, and coloring assignment $\sigma$ per root, parametrizes the Iwahori-fixed subspace.
• Equivalences: There exist weight-preserving bijections between colored Lusztig data, colored Gelfand–Tsetlin patterns, and colored lattice model states (“delta-ice”) for fixed boundary conditions (Naprienko, 2021).

The combinatorial formula for the Iwahori Whittaker function reads:

$\phi_w(\lambda, w'; z) = \sum_{(m, \sigma)\in \LLu(\lambda+\rho, w, w')} \prod_{\alpha>0} \mathcal{G}(m, \sigma, \alpha) z^{\sum_\alpha m_\alpha \alpha}$

with $\mathcal{G}(m,\sigma,\alpha)$ combining explicit Gauss sum and $q$-power terms.

This formalism refines McNamara's evaluation of metaplectic spherical Whittaker functions in terms of Lusztig data by introducing colorings and cell decompositions corresponding to generalized Mirković–Vilonen cycles in affine flag varieties.

7. Applications and Connections with Lattice Models, Duality, and Further Directions

Solvable lattice models parameterized by spectral data and global Drinfeld twists interpolate between Iwahori and metaplectic Whittaker functions, establishing a direct link between quantum group representation theory, lattice model combinatorics, and $p$-adic harmonic analysis (Brubaker et al., 2021). Partition functions are explicitly matched to Whittaker functions for various covers by selecting suitable twists.

Demazure operators act on these combinatorial data, yielding recurrence relations for Whittaker functions, with closed-form solutions available for ground states. Row transfer matrices realize half-vertex operators on $q$-Fock space modules, elucidating the vertex-algebraic structure underlying Whittaker models.

These frameworks collectively facilitate the study of formal degrees, Whittaker dimensions, and wavefront sets for genuine representations, highlighting profound connections between representation theory, algebraic combinatorics, and arithmetic duality phenomena. The approach lays the groundwork for a broad extension of the local Langlands correspondence in the field of covering groups, suggesting that "covering" phenomena are isolated in Whittaker and wavefront data rather than in formal degrees (Dong et al., 2023, Gao et al., 22 Jan 2026).