Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm looking at the induction of representations of a parabolic subgroup of $Sp_4$ into the whole group. There are some cases that the result is reducible, and I need to compute the dimensions of the subrepresentations. So I was wondering if there is a general procedure to compute the dimensions, like there is a pretty general procedure to check irreducibility - i.e. Mackey's criterion (which is how I found the cases that are reducible).

My question: if an induced representation is reducible, is there a relatively general method to compute the dimensions of the subrepresentations?

I should be able to do my specific example by reading the existing literature on $Sp_4$, but the articles I have read so far (B. Srinivasan, T. A. Springer) don't say how they figured out the dimensions.

share|cite|improve this question
Just a small precision : you're looking at complex representations of $\textrm{Sp}_4(k)$ with $k$ a finite field, right ? – Joel Cohen Jun 6 '11 at 23:49
yes indeed i am – simplequestions Jun 7 '11 at 6:21

The $G$-endomorphism ring of the induced representation $\mu$ is isomorphic to the convolution algebra of function on $G$ transforming from the right and the left by the conjugate of $\mu$. The double coset space $P\backslash G /P$ is computed by the Bruhat decomposition. In this fashion, you can find a basis $n = \sum_{x} n_x \dim(\rho_x)$ intertwiner $P_i$ with $P_i^2$ and $P_i P_j = 0$ for $i \neq j$. The dimension of the range is the dimension of the representations.

share|cite|improve this answer

Your Answer


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.