Harish-Chandra–Howe Coefficients

• Harish-Chandra–Howe coefficients are numerical invariants that quantify the expansion of characters and govern the transfer of representations in the Howe correspondence.
• They play a crucial role in linking dual pairs and Hecke algebra modules, ensuring unique bijections and minimal unipotent support in representation theory.
• Their application in local character expansions and harmonic analysis enables explicit, multiplicity-free computations for p-adic and finite reductive groups.

The Harish-Chandra–Howe coefficients are fundamental numerical invariants arising in the expansion of characters and the paper of correspondences between representations of reductive groups over local fields and finite fields. They appear in multiple settings: as coefficients in the local character expansion of admissible representations, as combinatorial integers governing the transfer of irreducibles under the Howe correspondence for groups over finite fields, and as transition elements relating bases in the Grothendieck groups of Hecke algebra modules. These coefficients encode deep connections between orbit theory, harmonic analysis, and the structure of unitary, symplectic, and orthogonal dual pairs, as well as their parahoric and Whittaker models.

1. Harish–Chandra Series, Dual Pairs, and Lusztig Parametrization

For a finite classical group $$G_m = \mathrm{U}_m(\mathbb{F}_q),\,\mathrm{Sp}_{2m}(\mathbb{F}_q)$$, or $O^\pm_{2m}(\mathbb{F}_q)$, a cuspidal pair $(L, \rho)$ consists of an $F_q$-Levi subgroup $$L \simeq \mathrm{GL}_{t_1}\times\cdots\times\mathrm{GL}_{t_r} \times G_{m - |\mathbf{t}|}$$ and a representation $$\rho = \sigma_1 \otimes \cdots \otimes \sigma_r \otimes \varphi$$ with $\sigma_i$ cuspidal for $\mathrm{GL}_{t_i}$ and $\varphi$ cuspidal for $G_{m - |\mathbf{t}|}$. The associated Harish–Chandra series is

$\E(G_m, \rho)_L = \left\{\,\pi \in \Irr(G_m)\mid \pi \subset R_L^{G_m}(\rho)\,\right\}.$

Lusztig partitions $\Irr(G)$ into Lusztig series $\mathcal{E}(G, (s))$ indexed by $F_q$-conjugacy classes of semisimple elements $s \in G^*$, with an explicit bijection

$$\mathfrak{L}_s^G : \mathcal{E}(G, (s)) \to \mathcal{E}(C_{G^*}(s), (1))$$

sending each representation to one with unipotent support. This parametrization is crucial for the transfer of representation-theoretic data across groups and for the reduction of the Howe correspondence to unipotent cases (Epequin, 2019).

2. Statement and Structure of the Howe Correspondence

Type I dual pairs over finite fields, such as $(\mathrm{U}_m(q), \mathrm{U}_{m'}(q))$ or $(\mathrm{Sp}_{2m}(q), \mathrm{O}_{m'}(q))$, play a central role in the Howe correspondence. Given the Weil representation $\omega_{m, m'}$ for $G_m \times G'_{m'}$, the induced correspondence on virtual characters

$$\Theta_{m, m'}: \mathcal{R}(G_m) \to \mathcal{R}(G'_{m'})$$

maps irreducible representations according to explicit branching laws determined by parabolic induction and restriction. The main theorem asserts that for $\pi$ in the Harish–Chandra series $$\mathcal{E}(G_m, \sigma \otimes \varphi)_{\mathbf{t}}$$, $\Theta_{m,m'}(\pi)$ is nonzero only if $m' \geq m'(\varphi)$, and every irreducible constituent falls into a single Harish–Chandra series $$\mathcal{E}(G'_{m'}, \sigma' \otimes \varphi')_{\mathbf{t}'}$$ (Epequin, 2020).

3. Definition and Computation of the Harish–Chandra–Howe Coefficients

The Harish–Chandra–Howe coefficients $c_{\pi, \pi'}$ are defined as the multiplicities in the expansion

$$\Theta_{m, m'}(\pi) = \sum_{\pi' \in \mathcal{E}(G'_{m'}, \rho')_{L'}} c_{\pi, \pi'} \pi'$$

where $c_{\pi, \pi'} = 1$ if $\pi' = \theta_{G, G'}(\pi)$ and $0$ otherwise, with $\theta_{G,G'}$ giving a bijection between parametrizing sets. Each Harish–Chandra series is multiplicity-free by results of Howlett–Lehrer and Geck–Hiss–Lübeck, leading to a combinatorially explicit rule on bipartitions: for unitary or symplectic–orthogonal dual pairs, $\theta$ acts by prescribed amalgamations of bipartitions or partitions, ensuring a unique nonzero coefficient for each source representation (Epequin, 2020).

Table: Key Structures Involved in Harish–Chandra–Howe Coefficients for Dual Pairs

Group $G$	Parametrization	Dual Group $G'$
$\mathrm{U}_m(q)$	Bipartitions	$\mathrm{U}_{m'}(q)$
$\mathrm{Sp}_{2m}(q)$	Bipartitions/Partitions	$\mathrm{O}_{m'}(q)$
$\mathrm{GL}_n(q)$	Partitions	$\mathrm{GL}_{n'}(q)$

The significance lies in the explicit and unitriangular nature of the correspondence, and in the fact that for each irreducible, there is precisely one image in the corresponding series, matching the unique nonzero coefficient.

4. Minimal Unipotent Support and Uniqueness

For each irreducible representation $\pi \in \Irr(G)$, Lusztig's theory assigns a unique unipotent support $\mathcal{O}_\pi$, corresponding to the maximal unipotent class on which $\pi$ does not vanish. Under the Lusztig correspondence, unipotent supports are preserved or become larger in the closure order. In the context of Howe correspondence, for each $\Theta_{G,G'}(\pi)$, there is a unique constituent with minimal unipotent support. This constituent is the one selected by the bijection $\theta_{G,G'}$ and is the one for which $c_{\pi,\pi'}=1$. On purely unipotent series, the explicit bijection on bipartitions aligns with this minimal-support rule (Epequin, 2020).

5. Harish–Chandra–Howe Coefficients in Local Character Expansion

For representations of $G_n = \mathrm{GL}_n(F)$ ($F$ a non-Archimedean local field), the local character expansion

$$\Theta_\pi(1 + X) = \sum_{\alpha \in P(n)} c_\alpha(\pi)\,\widehat{\mu}_{\mathcal{O}_\alpha}(X)$$

decomposes the character in terms of Fourier-transformed orbital integrals over nilpotent orbits $\mathcal{O}_\alpha$. The $c_\alpha(\pi)$ are the Harish–Chandra–Howe coefficients, which, for Iwahori-spherical representations, can be computed via the transition between two combinatorially-defined bases in the Grothendieck group of finite Iwahori–Hecke algebra modules. The change-of-basis matrix, constructed from counts of bipartite graphs with specified degree sequences, is unitriangular with ones on the diagonal, encoding an explicit relationship between the $c_\alpha(\pi)$ and principal degenerate Whittaker dimensions $d_\alpha(\pi)$ (Gurevich, 2022).

6. Applications and Implications

The Harish–Chandra–Howe coefficients are instrumental in performing fine-grained harmonic analysis on $p$-adic and finite reductive groups. They provide the bridge between character theory, parabolic induction, and the theory of automorphic forms and their Fourier coefficients. For dual pairs and the theta correspondence, they ensure the transfer is multiplicity-free and combinatorially computable, rendering the correspondence amenable to explicit calculations, including for degenerate Whittaker models and Iwahori–spherical representations. The coefficients also underlie the unique minimal support property, critical for classifying representations and understanding the structure of local and global correspondences (Epequin, 2020, Gurevich, 2022).

7. Interactions with Weak Cuspidality and Further Directions

Weak cuspidality provides a refined notion of cuspidal support, especially for complex or $\ell$-modular representations. The Howe correspondence respects this refinement: a weakly cuspidal representation only has a nonzero image at its first occurrence index, and at that index, the theta lift is irreducible and weakly cuspidal. The combinatorial rules for the coefficients remain valid in this broader framework, reinforcing the unifying role of the Harish–Chandra–Howe coefficients in bridging geometric, combinatorial, and analytic aspects of representation theory. Ongoing research leverages these structures for deeper investigations into local Langlands correspondences and harmonic analysis on $p$-adic and finite fields (Epequin, 2020).