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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.

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    $\begingroup$ Just a small precision : you're looking at complex representations of $\textrm{Sp}_4(k)$ with $k$ a finite field, right ? $\endgroup$
    – Joel Cohen
    Jun 6, 2011 at 23:49
  • $\begingroup$ yes indeed i am $\endgroup$
    – simplequestions
    Jun 7, 2011 at 6:21

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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.

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