rt.representation theory - How to think about parabolic induction.

I have also been trying to understand the big picture here. As far as I can tell, the key point is that two parabolic subgroups $P$ and $Q$ of $G$ have conjugate Levis if and only if $Pbackslash G/Q$ contains an orbit which is the preimage of a single Bruhat cell in $Pbackslash G$ and $G/Q$.

Some more details: (My apologies if I have missed the point of the question and the following is already well understood to everyone (or just wrong!). I have tried to write down a general argument that would apply in various settings, but I am probably missing important features of the proof in any particular setting.)

Suppose $P$ and $Q$ are parabolic subgroups of $G$ with Levi factors $L$ and $M$ respectively (I would also prefer to think about the Levis as being subquotients of $G$). I will not assume $L$ and $M$ are related to begin with.

Subsets of the double coset space $Pbackslash G/Q$ give rise to functors from representations of $L$ to representations of $M$. Taking the whole of $Pbackslash G/Q$ corresponds to the functor $Res_{M,Q}^G Ind_{L,P}^G$. Taking smaller subsets picks out summands of this (note that the Mackey formula expresses restriction then induction as a sum over such double cosets).

The parabolic $P$ gives rise to an embedding of the root systems of $L$ into the root system of $G$ (I am being slightly sloppy here about how/when I'm choosing Borels... this can be made more precise in a number of different ways, hopefully all equivalent) . The Weyl group $W_L$ gets idetified with a parabolic subgroup $W_P$ of the Weyl group $W$ (similarly for $(Q,M)$). The Bruhat decomposition idetifies $Pbackslash G/Q$ with $W_P backslash W/W_Q$ (as sets).

By a minor abuse of terminology Levis $L$ and $M$ can be said to be conjugate if the root system of $L$ is conjugate to the root system of $M$ by an element of $W$. If we had chosen embeddings of $L$ and $M$ as subgroups of $G$, this is equivalent to them being conjugate as subgroups (independantly of the choice of parabolic). This is also equivalent to $W_P$ being conjugate to $W_Q$.

It follows from the Bruhat decomposition that $L$ and $M$ are conjugate if and only if there is a double coset in $Qbackslash G/P$ which is the image of a single Bruhat cell in $Pbackslash G$ and $G/Q$.

Let me make things more concrete for a second: suppose we fix a Borel $B$, and assume $P$ and $Q$ contain $B$. Then $P$ and $Q$ are conjugate if and only if they are equal. In that case $P$ itself is a double coset in $Pbackslash G/P$. which is already a Bruhat cell (i.e. a point). If $P$ and $Q$ are not conjugate, but $W_P$ and $W_Q$ are, then there is an element $ain W$ such that $W_PaW_Q = W_Pa = aW_Q$. This means that $PaQ = BaQ = PaB$.

Hence there is a canonical functor from $Rep(L)$ to $Rep(M)$. This functor is invertible (its inverse is the corresponding double coset in $Qbackslash G/P$). If we identify $L$ and $M$ compatibly with the identification of root systems then I claim this functor is the identity. Moreover, by its construction, this functor is a summand of $Res_{M,Q}^G Ind_{L,P}^G$. For a representation $V$ of $L=M$, the identification of $Ind_{L,P}^G$ with $Ind_{L,Q}^G$ is the adjoint of the inclusion $Vto Res_{L,P}^GInd_{L,Q}^GV$.

Remarks:
The parabolic induction and restriction functors can be thought of as pull-push of sheaves along the following correspondence:

$BL leftarrow BP rightarrow BG$.

The composition of the correspondences associated to $(P,L)$ and $(Q,M)$ is

$BL leftarrow BP times_{BG} BQ = Pbackslash G/Q rightarrow BM$.

For the anologue of these ideas in the setting of character sheaves, one should replace $Pbackslash G/Q$ with its loopspace, the corresponding relative Steinberg variety.