Computing the dimensions of representations in a reducible induced representation

Question

This is a question on math.se that got no answers.

1) Is there a relatively general method of computing the dimensions of representations in a reducible induced representation?

An explicit specific example is: $G=Sp_4(\mathbb{F}_q)$, $J$ the antidiagonal $(1,1,-1,-1)$, $P$ the parabolic corresponding to $4=2+2$, $\tau$ a representation of $P$ that comes from a representation of $GL_2(\mathbb{F}_q)$, which will also be denoted $\tau_0$ (the matrix $\[\[g,\*\],[0,\*\]\]$ acts as $\tau_0(g)$). Mackey's irreducibility criterion shows that $Ind_P^G \tau$ is reducible iff the central character $\omega_{\tau_0} =1$. How does one compute the dimensions of the (two) subrepresentations in this case? Note that this specific case is solved: "The characters of the finite symplectic group $Sp(4,q)$", B. Srinivasan, but the computations there are quite elaborate and not in the spirit of the question.

In the above example a short attempt to use Deligne-Lusztig theory seems to fail since the irreducible subrepresentations seem to be geometrically conjugate. Maybe the short attempt is ridiculous.

2) Is there a way to compute the dimensions, or even the characters themselves, using Deligne-Lusztig theory? (i.e. same question only restricted to reductive groups over finite fields)

Answer 1

Certainly for the finite groups of Lie type there is a general technology for treating induced characters from parabolics, based on Deligne-Lusztig theory. Even without getting fully into that story, Lusztig's methods yield (recursively) the degrees of irreducible characters. The 1985 book by Roger Carter Finite Groups of Lie Type develops most of the theory up to that time, though later papers by Lusztig and others refine the results in many directions. See for example Chapter 10 of Carter's book for the decomposition of induced characters in the Hecke algebra spirit. (Some of this was worked out earlier by people including Curtis, Iwahori, Kilmoyer in the important special case of the character induced from the trivial character of a Borel subgroup in the finite group, or parabolic analogues.)

Small rank groups had been treated earlier in some detail, as noted in the question for $Sp_4$, but the general methods lead much farther than the ad hoc methods used in early papers.

It should be emphasized that general methods like Mackey theory for finite groups stop well short of solving these kinds of problems for groups of Lie type. Even the original Deligne-Lusztig paper has to be supplemented by further work on Hecke algebra methods for decomposition of induced characters. But eventually this all becomes a unified theory for these particular families of groups. (A short treatment by Digne-Michel is also given in their LMS Student Text softcover book, but with less concrete detail than Carter gives.)