Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^n$. Let $\mathrm{B}$ be the upper triangular matrices. Consider a character $\mu : \mathrm{B}(R_n)\rightarrow \mathbb{C}^\times$. When is the induced representation $ \mathrm{Ind}_{\mathrm{B}(R_n)}^{\mathrm{GL}_2(R_n)} \mu$ irreducible?

share|improve this question
1  
I guess the exponent $r$ here should be $n$? –  Jim Humphreys Mar 23 '11 at 15:44
add comment

2 Answers 2

up vote 4 down vote accepted

A sufficient criterion for irreducibility is given, for example, in Theorem 4.6 in Hill: Semisimple and cuspidal characters of $\mathrm{GL}_n(\mathcal{O})$. Hill's result is more general, and holds for certain representations of $\mathrm{GL}_n(\mathcal{O})$, for $n\geq 2$. For $\mathrm{GL}_2(\mathcal{O})$ it says the following. Let $T$ be the diagonal torus, so that $T(\mathcal{O}_r)\cong\mathcal{O}_r^{\times}\times\mathcal{O}_r^{\times}$. Let $\theta=\theta_1\theta_2$ be a character of $T(\mathcal{O}_r)$, where $\theta_1$ and $\theta_2$ are characters of $\mathcal{O}_r^\times$. Suppose that the restriction of $\theta_1$ to $1+\mathfrak{p}^{r-1}$ is non-trivial, and that the restriction of $\theta_1$ to $1+\mathfrak{p}^{r-1}$ is not equal to that of $\theta_2$. Let $\tilde{\theta}$ denote the pull-back of $\theta$ to $B(\mathcal{O}_r)$. Then the representation $$\mathrm{Ind}^{\mathrm{GL}_2(\mathcal{O}_r)}_{B(\mathcal{O}_r)}\tilde{\theta}$$ is irreducible.

share|improve this answer
add comment

For PGL(2) I have the following reference :

Silberger, A.J. : PGL 2 over the p-adics : its representations, spherical functions, and Fourier analysis. Springer Lecture Notes in Mathematics 166. Berlin, Heidelberg, New York: Springer 1970

I don't have this book with me, but the basic idea is to apply Mackey's irreducibility criterion. For this you have to determine the double cosets of $G={\rm GL}_2 (R_u )$ mod $B=B(R_u )$. For $n=1$, just use the Bruhat decomposition. To get a set of representatives of the double quotient for $n>1$, you introduce the "Iwahori" subgroup $I$ of matrices that are upper triangular mod ${\mathfrak p}$. Then $G=I\cup IwI$, where $w$ is the standard Weyl group element. Next you have the following set of representatives of right (or left) $B$-cosets in $I$ : the lower triangular unipotent matrices with the coefficient varying over ${\mathfrak p}/{\mathfrak p}^{n}$.

share|improve this answer
1  
In the end, I think the criterion would be that $\mu^{w}$ is not isomorphic to $\mu$. –  Joël Cohen Mar 24 '11 at 13:56
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.