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)

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)