2
$\begingroup$

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}_p$ with maximal split torus $T_s$, Borel subgroup $B = TN$ and Weyl group $W(G,T_s)$. Let us consider the $\mathbb{Q}_p-$points of $G$ and $B = TN$ that we will also denote by $G$ and $B = TN$. Given $\xi:\;T\to\mathbb{C}^{\times}$ an unramified character we define the normalized induction $\mathrm{Ind}_{B}^{G}\xi$ as the space of smooth functions $f:\;G\to\mathbb{C}$ such that $f(bgk) = \delta^{1/2}_B(b)\xi(b)f(g)$. For regular characters, i.e. $\xi$ such that $\mathrm{Stab}_{W(G,T_s)}\xi = id$, Casselman gave a if only if criterion for the irreducibility of $\mathrm{Ind}_{B}^{G}\xi$ in prop. 3.5, (b), p. 401, "The unramified principal series of p-adic groups. I. The spherical function". Is there an equivalent result when we remove the hypothesis of the regular character?

If we suppose that the character $\xi$ is unitary, we know that $\xi$ regular implies that $\mathrm{Ind}_{B}^{G}\xi$ is irreducible. Are there results in the other direction?

Improve this question
$\endgroup$
10
  • $\begingroup$ By "a topological reductive group" I think you actually mean "a reductive algebraic group over a nonarchimedean local field". $\endgroup$
    – David Loeffler
    Commented Jun 16, 2022 at 10:02
  • $\begingroup$ By the way, how exactly does $W(G, T_s)$ act on characters of $T$ if $T_s \ne T$? $\endgroup$
    – David Loeffler
    Commented Jun 16, 2022 at 10:05
  • $\begingroup$ Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. The group $W(G,T_s) \simeq N_{G}(T_s)/T$. The unramified characters satisfy that $\mathrm{Hom}(T(\mathbb{Q}_p)/T(\mathbb{Z}_p),\mathbb{C}^{\times}) \simeq X^{\bullet}(T)\otimes_{\mathbb{Z}}\mathbb{C}$, where $X^{\bullet}(T)$ are the rationals characters of the algebraic torus $T$. Given $\xi$ an unramified character of $T(\mathbb{Q}_p)$ and $w\in W(G,T_s)$, then $w\cdot\xi(\cdot) = \xi(x^{-1}\cdot x)$ where $x$ is a representative for $w$ in the quotient $N_{G}(T_s)/T$. $\endgroup$
    – Aersk
    Commented Jun 16, 2022 at 11:13
  • $\begingroup$ Thank you for your comment about the reductive group, I will correct my question. $\endgroup$
    – Aersk
    Commented Jun 16, 2022 at 11:15
  • $\begingroup$ Your definition of $W(G, T_s)$ seems to depend on $T$ (which is not part of the data); I think $N_G(T_s) / Z_G(T_s)$ is the usual definition. For your definition of the action on characters, you seem to be implicitly assuming that $w$ has a representative that normalises $T$ (not just $T_s$); and I don't see anything in the setup that guarantees this. $\endgroup$
    – David Loeffler
    Commented Jun 16, 2022 at 11:20

0

Reset to default

You must log in to answer this question.