Can we compare $K$-spherical representations of $p$-adic groups for varying special maximal subgroups $K$?

Question

In this question, I'm borrowing the notations from Minguez' paper on unramified representations of unitary groups. Let $F$ be a $p$-adic field and let $G$ be a connected reductive group over $F$. Let $P_0$ be a minimal $F$-parabolic subgroup and $M_0$ a Levi factor of $P_0$ defined over $F$. Let $W := \mathrm{N}_G(M_0)/M_0$ denote the spherical Weyl group of $G$. We denote by $X^{\mathrm{un}}(M_0)$ the set of unramified characters of $M_0$, that is those (complex) characters of $M_0$ which are trivial on the unique maximal compact subgroup of $M_0$. The Weyl group $W$ acts on $X^{\mathrm{un}}(M_0)$ through the formula $(w\chi)(m) := \chi(w^{-1}mw)$.

We give ourselves a good special maximal compact subgroup $K$ in $G$. A smooth representation of $G$ is called $K$-spherical if it contains a non-zero $K$-invariant vector. Then the set of isomorphism classes of $K$-spherical representations is in bijection with the quotient set $X^{\mathrm{un}}(M_0)/W$.

Such a bijection is induced by the following map $X^{\mathrm{un}}(M_0) \rightarrow \{K\text{-spherical representations}\}$. Let $\chi \in X^{\mathrm{un}}(M_0)$ and denote by $\iota_{P_0}^G\,\chi$ the normalized parabolic induction of $\chi$. The space of $K$-invariant vectors in $\iota_{P_0}^G\,\chi$ has dimension $1$ and therefore, there is a unique composition series $\pi_{K,\chi}$ of $\iota_{P_0}^G\,\chi$ which is $K$-spherical. The map is then $\chi \mapsto \pi_{K,\chi}$.

My question is about what happens if I allow $K$ to vary among all good special maximal compact subgroups of $G$. For instance, if $G$ is a unitary group then there are two distinct conjugacy classes of such subgroups.

If $K'$ is another such subgroup, are $K$-spherical and $K'$-spherical representations the same thing ? In other words, are the representations $\pi_{K,\chi}$ and $\pi_{K',\chi}$ the same ? Or is there a concrete counterexample ?

Unless I'm mistaken, if $\pi$ is a character of $G$ then it is $K$-spherical if and only if it is $K'$-spherical ; at least it seems to be the case for unitary groups. But what about higher dimensional representations ?

Answer 1

If I'm not mistaken, $\pi_{K,\chi}$ and $\pi_{K',\chi}$ can be different.

Consider a bipartite bi-regular tree, with degrees $q_1+1<q_2+1$. There are $p$-adic unitary groups that their building in such a tree with $q_2=q_1^3$, see for example https://arxiv.org/abs/1005.3504.

Then there are two maximal compact subgroups, corresponding to the stabilizers of the two types of vertices, and I believe that they are both special. Denote them by $K_1$ and $K_2$.

Now, if you look at Proposition 3.2 in https://arxiv.org/abs/1005.3504, there is a classification of the representations of the corresponding Iwahori-Hecke algebra, which are essentially the same as the possible subquotients of $\iota_{P_0}^G\chi$ in the question, and is denoted $X(\nu)$ in that paper. In this case there are two interesting representations that do not occur in the regular case, which are denoted by $\operatorname{ds}$ and $\operatorname{nt}$ in the said proposition, and are the subquotients of some $\iota_{P_0}^G\chi$ for some specific value of $\chi$. It holds that $\operatorname{ds}= \pi_{K_1,\chi}$ while $\operatorname{nt}= \pi_{K_2,\chi}$.

The reason is that the Iwahori-Hecke algebra is generated by two elements $T_1,T_2$, which satisfy the formula $(T_i+1)(T_i-q_i)=0$. The $K_i$-fixed subspace is the same as the $q_i$-eigenspace of $T_i$, and you may check that on $\operatorname{ds}$ the operator $T_1$ acts by $q_1$ and $T_2$ acts by $(-1)$, while on $\operatorname{nt}$ the operator $T_1$ acts by $(-1)$ and $T_2$ acts by $q_2$.

It is also related to the fact that on biregular (and non-regular) trees there is a spherical eigenfunction of the adjacency operator of eigenvalue 0, which is in $L^2$, and hence a discrete series (this is where "ds" comes from). It is supported only on vertices of degree $q_1+1$. Similarly, there is a "non-tempered" ("nt") eigenfunction of eigenvalue 0, supported on vertices of degree $q_2+1$.

On the other hand, there is only one $\chi$ where this situation happens, so you can expect it to be quite rare in general. Two special maximal compact subgroups are usually conjugate under an adjoint action (see Tits, "Reductive groups over local fields", Section 2.5), and I don't think that this property can happen in this case.