A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fields

TL;DR

This paper constructs a canonical way to decompose irreducible representations of the multiplicative group of a division algebra over a local field into one-dimensional projective bases, advancing understanding of their structure.

Contribution

It introduces a method to explicitly construct projective bases for irreducible representations of these groups, a novel approach in the representation theory of division algebras over local fields.

Findings

Canonical decomposition of irreducible representations into one-dimensional subspaces

Explicit construction of projective bases for these representations

Enhanced understanding of the structure of multiplicative groups of division algebras

Abstract

Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $ G=D^{\star}$ be the multiplicative group of $D$. The goal of this paper is to provide a canonical decomposition of any complex irreducible representation $V$ of $G$ in a direct sum of one-dimensional subspaces.

Equations

$$\kappa :G_0\times K^1\times \underline{G}(F)\to \underline{G}(\mathbb{A})$$

$$(D_0^{\ast }\times K^1)\cap G_E=\{e\}$$

$$f\star \alpha (a))=\hat{a}(f),a\in \mathcal{A},f\in {\mathbb{C}}_c(G_0)$$

$$V={\oplus }_{\chi \in {\Xi }_V}{V}_{\chi }$$

Full text

David Kazhdan

Let $k=\mathbb{F}_{q}$ be a finite field, $F:=k(t)$ the field of rational functions on the projective line $\underline{\mathbb{P}}^{1}$ over $k$. Let $S$ be the set of points of $\underline{\mathbb{P}}^{1}$. For any point $s\in S$ we denote by $F_{s}$ the completion of $F$ at $s$, by $\nu_{s}:F_{s}\to{\mathbb{Z}}\cup+\infty$ the valuation map and by $\mathcal{O}_{s}\subset F_{s}$ the subring of integers. We denote by $\mathbb{A}$ the ring of adeles for $F$.

Let $D$ be a skew-field with center $F$ unramified outside $\{0,{\infty}\},D_{0}:=D\otimes_{F}F_{0}$ and $D_{\infty}:=D\otimes_{F}F_{\infty}$. We have $dim_{F}(D)=n^{2}$.

We denote by $\underline{G}$ the mutiplicative group of $D$ considered as an algebraic $F$-group, write $G_{s}:=\underline{G}(F_{s})$ and denote by ${\mathbb{C}}_{c}G_{0})\subset{\mathbb{C}}(G_{0})$ the algebra of locally constant compactly supported functions on $G_{0}$.

For any point $s\in S,s\neq 0,\infty$ we identify the group the group $\underline{G}(F_{s})$ with $Gl_{n}(F_{s})$ and define $K_{s}:=Gl_{n}(\mathcal{O}_{s})$.

Let $N_{\infty}:G_{\infty}\to F_{\infty}^{\star}$ be the reduced norm. We define $K\infty=\{g\in D_{\infty}|\nu_{\infty}(N_{\infty}(g))\geq 0\}$ and $K^{1}_{\infty}=\{g\in D_{\infty}|\nu_{\infty}(N_{\infty}(g-1))>0\}$. Then $K_{\infty}\subset G_{\infty}$ is an open compact subgroup and $K_{\infty}/K^{1}_{\infty}=\mathbb{F}_{q^{n}}^{\star}$. We define $K^{1}:=\prod_{s\in S-\{0,\infty\}}K_{s}\times K^{1}_{\infty}$.

The multiplication defines a map

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This paper is based on the following result

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which is obvious.

∎

Let $R$ be the space of ${\mathbb{C}}$-valued locally constant functions on $\underline{G}(\mathbb{A})/\underline{G}(F)K^{1}$, $\mathcal{H}_{s},s\neq 0,\infty$ be the spheical Hecke algebra at $s,\mathcal{H}:=\prod_{s\in S-\{0,\infty\}}\mathcal{H}_{s}$.

We have a natural action $a\to\hat{a}$ of the commutative algebra $\mathcal{A}:=\mathcal{H}\otimes{\mathbb{C}}[K^{0}_{\infty}/K^{1}_{\infty}]$ on $R$.

1. (1)

The natural action of the group $G_{0}$ on the space $X:=\underline{G}(\mathbb{A})/K^{1}\times\underline{G}(F)$ is simply transitive. So we can identify $X$ with $G_{0}$. 2. (2)

The restriction to $G_{0}$ defines a $G_{0}$-equivariant isomorphism $u:R\to{\mathbb{C}}(G_{0})$. 3. (3)

For any irreducible representation $V$ of $G_{0}$ the restriction to $G_{0}$ defines an isomorphism $u_{V}:Hom_{G_{0}}(V^{\vee},L)\to V$ where $V^{\vee}$ is the representation dual to $V$. 4. (4)

There exists a map $\alpha:\mathcal{A}\to{\mathbb{C}}_{c}(G_{0})$ such that

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Let $\Xi$ be the set of homomorphisms $\chi:\mathcal{H}\otimes{\mathbb{C}}[\mathbb{F}_{q^{n}}^{\star}]\to{\mathbb{C}}$. For any $\chi\in\Xi$ we define $V_{\chi}:=\{v\in V|av=\chi(a)v\}$ for all $a\in\mathcal{H}\otimes{\mathbb{C}}[\mathbb{F}_{q^{n}}^{\star}]$. Let $\Xi_{V}=\{\chi\in\Xi|V_{\chi}\neq\{0\}\}$.

1. (1)

$dim(V_{\chi})=1$* for all $\chi\in\Xi_{V}$.* 2. (2)

$V=\oplus_{\chi\in\Xi_{V}}V_{\chi}$.

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where the subspaces $V_{\chi}\subset V$ are $\mathcal{H}\times G_{\infty}$-invariant and the representation $\tilde{\rho}_{\chi}$ of $\mathcal{H}\times G_{\infty}$ on $V_{\chi}$ is irreducible. Since $\mathcal{H}$ is commutative this implies the irreduciblity of the restriction $\rho_{\chi}$ of $\tilde{\rho}_{\chi}$ to $G_{\infty}$. By definition we can consider $\rho_{\chi}$ as a representation of the quotient group $G_{\infty}/K^{1}_{\infty}={\mathbb{Z}}\ltimes\mathbb{F}^{\star}_{q^{n}}$ where $1\in{\mathbb{Z}}$ acts by the Frobenious automorphism on $\mathbb{F}^{\star}_{q^{n}}$. It is easy to see that the restriction of any irreducible representation of the group ${\mathbb{Z}}\ltimes\mathbb{F}^{\star}_{q^{n}}$ on $\mathbb{F}^{\star}_{q^{n}}$ is the direct sum of distinct one-dimensional representations. ∎

Acknowledgments. The project has received funding from ERC under grant agreement 669655.