Harish-Chandra Character Expansion

• Harish-Chandra character expansion is a method that expresses a representation's character as a linear combination of Fourier transforms of invariant orbital integrals on nilpotent orbits.
• The expansion coefficients, determined via degenerate Whittaker models or Kazhdan–Lusztig polynomials, capture essential representation-theoretic invariants.
• This framework extends to supercuspidal and principal series representations, linking harmonic analysis, orbital geometry, and combinatorial structures in reductive p-adic groups.

The Harish-Chandra character expansion is a foundational tool in $p$-adic representation theory, yielding precise local expansions of distribution characters of admissible representations in terms of explicit geometric and combinatorial data. It expresses the character of a representation as a linear combination of Fourier transforms of invariant orbital integrals associated to nilpotent orbits on the Lie algebra, providing a deep interaction between harmonic analysis, the geometry of orbits, and the structure of representations of reductive groups over local fields.

1. Definition and Structure of the Local Character Expansion

Let $G$ be a reductive $p$-adic group, such as $G_n = \GL_n(F)$, and $\mathfrak{g}_n = \mathrm{Mat}_{n \times n}(F)$ its Lie algebra. For a smooth admissible representation $\pi$ of $G$, the (distributional) character $\Theta_\pi$ is represented by a locally integrable function on the set of regular elements. The Harish-Chandra–Howe local character expansion asserts that, in a sufficiently small neighborhood $U$ of the identity, $\Theta_\pi$ admits an explicit expansion: $$\Theta_\pi(1 + X) = \sum_{\alpha \in P(n)} c_\alpha(\pi)\, \widehat\mu_{\mathcal{O}_\alpha}(X), \quad X \in \mathfrak{g}_n \text{ small regular},$$ where $P(n)$ denotes the set of partitions of $n$, each $\mathcal{O}_\alpha$ is the unique nilpotent orbit of Jordan type $\alpha$, $\widehat\mu_{\mathcal{O}_\alpha}$ is the Fourier transform of the invariant orbital integral on $\mathcal{O}_\alpha$, and $c_\alpha(\pi) \in \mathbb{Z}$ are the local expansion coefficients (Gurevich, 2024, Gurevich, 2022).

These coefficients encapsulate significant representation-theoretic invariants and are directly computable for large classes of representations, such as principal series and supercuspidal representations.

2. Invariant Orbital Integrals and Their Fourier Transforms

Fundamental to the construction of the expansion are the orbital integrals on the Lie algebra. For each nilpotent adjoint $G$-orbit $\mathcal{O}$ and a compactly supported smooth function $f$ on $\mathfrak{g}$,

$$\mu_{\mathcal{O}}(f) = \int_{G/C_G(X)} f(g^{-1} X g)\, dg, \quad X \in \mathcal{O},$$

where $C_G(X)$ is the centralizer of $X$. The Fourier transform $\widehat \mu_{\mathcal{O}}(X)$ is defined by evaluating the orbital integral against the exponential function $\exp(\mathrm{tr}(\cdot\, X))$. This normalization ensures that $\widehat \mu_{\mathcal{O}}(X)$ is locally constant on the set of regular semisimple $X$ near zero and provides the analytic building blocks for the expansion (Gurevich, 2024, Spice, 2010).

In low rank, explicit formulae for $\widehat \mu_{\mathcal{O}}$ are computable in terms of Gauss sums, Jacobians, and residue field invariants. For example, for $G = \SL_2$, each nilpotent orbit corresponds to a closed-form germ involving discriminant and local sign data (Spice, 2010).

3. Character Expansion Coefficients and Degenerate Whittaker Models

For the principal series representations of $\GL_n(F)$, the coefficients $c_\alpha(\pi)$ in the expansion admit two explicit descriptions. The first formula relates $c_\alpha(\pi)$ to the dimensions of degenerate Whittaker models associated with relevant Jacquet functors: $$d_\alpha(\pi) = \dim_\mathbb{C} Wh\left( \mathbf{r}_\alpha (\pi) \right),\quad d_\alpha(\pi) = \sum_{\beta \in P(n)} s(\alpha, \beta^t) c_\beta(\pi),$$ where $s(\alpha, \beta^t)$ counts $0$–$1$ matrices with given row and column sums, and $\mathbf r_\alpha$ is the normalized Jacquet functor for partition $\alpha$ (Gurevich, 2024).

This formula yields a unitriangular system (with respect to dominance order) for recovering $c_\beta(\pi)$ from $d_\alpha(\pi)$. For Iwahori-spherical representations, this system is realized at the level of modules for the finite Iwahori-Hecke algebra, with Zelevinsky’s PSH algebra providing the combinatorial framework (Gurevich, 2022).

4. Kazhdan–Lusztig Polynomial Formula for Principal Series

The second explicit description applies to integral principal series, where the coefficients $c_\mu(\pi)$ are given in terms of Kazhdan-Lusztig polynomials associated to suitable symmetric groups: $$c_{\mu}(\pi) = \sum_{\substack{ \tau \in S_k \ p(\tau*_{-\lambda} \alpha) = \mu^t }} \epsilon(\tau)\, P_{\omega_0 \omega_1 \tau \omega_2,\, \omega_0 \omega_1 \omega_2}(1),$$ where $P_{x, y}(q)$ is the Kazhdan–Lusztig polynomial, $p(\cdot)$ arranges the composition into nonincreasing order, and the sum is over elements of $S_k$ compatible with the partition $\mu$. This approach uses the Zelevinsky classification of integral representations, with the basis change from standards to irreducibles governed by the KL polynomials (Gurevich, 2024).

This formula enables the computation of $c_\mu(\pi)$ purely via combinatorial and Hecke-theoretic data without recourse to Whittaker model dimensions.

5. Generalization to Supercuspidals and Recursion via Gauss Sums

For tame supercuspidal representations, the local character expansion is extended to expansions about arbitrary semisimple elements $\gamma$: $$\Theta_\pi\bigl(\gamma\,\exp(H)\bigr) = \sum_{\mathcal{O} \subset \mathrm{Nil}(\mathfrak{g}_\gamma)} c_{\mathcal{O},\gamma}(\pi)\, \widehat \mu_{\mathcal{O}}(H),$$ with each $\mathcal{O}$ a nilpotent orbit in $\mathrm{Lie}(\mathrm{Cent}_G(\gamma))$. The coefficients $c_{\mathcal{O},\gamma}(\pi)$ are computed recursively by descending through an explicit sequence of twisted Levi subgroups. At each induction step, a "fourth-root-of-unity" Gauss sum factor, corresponding to the Weil index of a naturally associated quadratic form, is introduced (Spice, 2017, Spice, 2021).

At depth zero, the expansion reduces to the standard Howe–Harish-Chandra form. For positive-depth representations, the recursion structure expresses $c_{\mathcal{O},\gamma}(\pi)$ in terms of coefficients for representations on lower-rank Levi subgroups, interpolated by explicit Gauss sum ratios.

6. Connections with Hecke Algebras, Hopf Structures, and Examples

In the Iwahori–spherical and depth-zero case, the expansion coefficients $c_\alpha(\pi)$ coincide with the coordinates of the finite Hecke algebra module $\pi^{I_n}$ in the trivial-induction basis, and degenerate Whittaker model dimensions are realized as pairings in the Zelevinsky PSH algebra. The unitriangular transition matrix $T_{\alpha,\beta} = s(\alpha, \beta^t)$ encodes the change of basis and is tightly controlled via combinatorics of bipartite graphs; inversion of this system allows recovery of $c_\beta(\pi)$ from $d_\alpha(\pi)$ (Gurevich, 2022).

For $\GL_2$ and $\GL_3$, explicit matrices illustrate the transition: | $n$ | Partitions | $T$-matrix entries $s(\alpha, \beta^t)$ | |-----|------------|-----------------------------------------| | 2 | (2), (1,1) | $$T_2 = \begin{pmatrix} 1 & 0 \ 2 & 1 \end{pmatrix}$$ | | 3 | (3), (2,1), (1,1,1) | $$T_3 = \begin{pmatrix} 1 & 0 & 0 \ 3 & 1 & 0 \ 3 & 2 & 1 \end{pmatrix}$$ |

For the generic principal series, $d_{(n)} = 1$, $d_{\alpha} = 0$ for $\alpha \neq (n)$, yielding $c_{(n)}=1$, $c_{\text{others}}=0$. For the Steinberg representation, the wavefront orbit is regular, with the only nonzero $c_\alpha(\pi)$ for the regular nilpotent orbit (Gurevich, 2024, Gurevich, 2022).

7. Endoscopy, Stability, and Further Applications

The recursive expansion framework generalizes to questions of endoscopic transfer and stability. The Gauss sum factors appearing in the inductive formulas coincide with the transfer factor fourth roots of unity (Kottwitz–Shelstad), enabling term-by-term matching of expansions and affirming stability of the resulting expansions under endoscopic lifting (Spice, 2017).

Applications include explicit construction of $L$-packets, computation of matching measures on "good" neighborhoods defined by Moy-Prasad filtrations, and reduction of analytic questions in harmonic analysis of $p$-adic groups to finitely computable combinatorics and explicit orbital integrals.

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The Harish-Chandra character expansion thus provides an explicit, functorial, and combinatorially transparent link between local harmonic analysis, orbit geometry, and deep structural invariants of admissible representations, with precise formulas now available for broad classes such as all principal series of $\GL_n$ (Gurevich, 2024, Gurevich, 2022), and all depth-zero or tame supercuspidals in positive characteristic settings (Spice, 2017, Spice, 2021, Spice, 2010).