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I'm a beginner in Bruhat-Tits theory, and the following phenomenon makes me puzzled.

So let $F$ be a $p$-adic field, with $\mathfrak{o}\supset \mathfrak{p}$ its ring of integers and maximal ideal, and $q$ its residue field order. Let $G$ be a connected reductive group over $F$. We assume $G$ to be quasi-split, and fix a Borel $B=TU$, $U$ the unipotent radical of $B$ and $T$ a maximal torus, all over $F$. Then we let $\Sigma=\Sigma(G,T_0)\subset X^*(T_0)_F$ be the root system ($T_0$ the split component of $T$).

Let $P$ be a standard $F$-parabolic subgroup (i.e. $P\supset B$), with its Levi decomposition $P=MN$ ($M$ a Levi and $N$ the unipotent radical), and $\bar{P}$ its opposite parabolic. Then we write $\Sigma(\bar{P})$ (resp. $\Sigma_{\mathrm{red}}(\bar{P})$) for the set of roots (resp. reduced roots) appearing in the Lie algebra $\bar{\mathfrak{n}}:=\mathrm{Lie}(\bar{N})$.

Let $I_G$ be a "standard" Iwahori subgroup (I'm sorry that I have not yet understood the precise definition here, but roughly, $I_G$ should "match" $B$) with pro-unipotent radical $I_G^+$ (a notion which so far I didn't fully understand either; nor did I find a precise definition in the literature, for example, in Kaletha-Prasad's famous new book), and $K_G$ a special maximal compact subgroup of $G(F)$ containing $I_G$. The simplest example is: $G=\mathrm{GL}(n)$ and $B$ the standard upper triangular Borel. Then $K_G=\mathrm{GL}(n,\mathfrak{o})$, $I_G$ is $$ \begin{bmatrix} \mathfrak{o} & \dots & \mathfrak{o}\\ & \ddots & \\ \mathfrak{p} & & \mathfrak{o} \end{bmatrix}$$ and $I_G^+$ is $$ \begin{bmatrix} 1+\mathfrak{p} & \dots & \mathfrak{o}\\ & \ddots & \\ \mathfrak{p} & & 1+\mathfrak{p} \end{bmatrix}.$$ In this example it's very clear, and reasonable, that $$(*) \prod_{\alpha\in \Sigma_{\mathrm{red}}(\bar{P})} [U_\alpha\cap K_G:U_\alpha\cap I_G^+]^{-1} = q^{-\dim \bar{\mathfrak{n}}}$$ for any standard parabolic $P$, where $U_{\alpha}$ is the root subgroup associated to $\alpha$. After all, the pro-unipotent radical $I_G^+$ should "contain some $\mathfrak{p}$ at negative root subgroups", and $K_G$ should "contain some $\mathfrak{o}$".

However, some "strange" phenomenon appears when the group is not split. There are examples of non-split, quasi-split $G$ with $P=MN$ such that $$\quad \prod_{\alpha\in \Sigma_{\mathrm{red}}(\bar{P})} [U_\alpha\cap K_G:U_\alpha\cap I_G^+]^{-1} \neq q^{-\dim \bar{\mathfrak{n}}}.$$ This looks somewhat weird. Why would such phenomena exist? This seems to mean that the pro-unipotent radical in BT-theory is very different from the "usual" unipotent radical. How do experts think about this?

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1 Answer 1

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A reference for the "pro-unipotent radical" of a parahoric subgroup is given in

Moy, Allen; Prasad, Gopal; (1994). "Unrefined minimal K-types for p -adic groups." Inventiones Mathematicae 116(1): 393-408.

The definition is given at page 387. The authors use the terminology "pro-nil radical" instead of "pro-unipotent radical", but this is the same object.

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