# On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this.

Let $F$ be the local field whether p-adic or archimedian. Let $V$ be a $n$-dimensional quadratic vector space over $F$ and $V^-$ the same space $V$ but opposite quadratic sign to $V$. Let $G:=GL(V)$ and $G^\diamond$:=$GL(V\oplus V^-)$.Let also $i: G \times G \hookrightarrow G^\diamond$ be the inclusion map.

Let $P$ be a Siegel-parabolic subgroup of $G^\diamond$ stabilizing $V^{\triangle}:=((x,x)\in V \oplus V^-)$ with Levi-component $GL(V^{\triangle})$ and $K$ a maximal compact subgroup of $G^\diamond$ such that $G^\diamond=PK$.

We define some $K$-invariant function $F$ on $G^\diamond$ as follows;

For $\tilde{g}=pk\in G^\diamond=PK$, define

$F(\tilde{g}):=|\det(p)|$ (here, the determinants are taken with respect to $GL(V^{\triangle})$ which is isomorphic to the Levi of $P$.)

Then we can define a function on $G$ given by $f(g):=F(i(g,1))$.

When $G$ is not $GL(n)$ but $O(V)$ or $Sp(V)$, it is explicitly described for the diagonal in proposition 6.4 in the book 'L-functions for the Classical group' written by Gelbart, Piatetski-Shapiro, Rallis. But they didn't say any about $GL(n)$.

So, I am wondering what is the exact expression in the $GL(n)$ case.

Especially, when $n=2$ and $g$ is given by

\begin{pmatrix} x & 0 \\\ 0 & y \end{pmatrix} satisfying $|x|\le |y|$, I guess the above $f(g)$ should be $(max(1,|xy|,|y|))^{-1}$ for $F$ is a $p$-adic case and $((1+x^2)(1+y^2))^\frac{-1}{2}$ for real field $F$. Is it right?

Since I am very curious whether my formula is right, I will be very thankful if someone give me some brightness on this.

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For me a height function associated to a parabolic subgroup is the modular character of this subgroup extended to the whole group by a choice of an Iwaswa decomposition. Eg. Sl(2) has two Iwasawa decompositions associated to the Borel. –  Marc Palm Apr 3 '13 at 16:38
In your case, similiar to Gl(n), there is probably only one choice. Do you mean the usual height function as described by me? –  Marc Palm Apr 3 '13 at 16:46
Palm,s I am sorry! I mean different height function that denotes $f(g)$ in my post. Here, the name of height function is not the usual height function, but used only to describe the function in my question. –  James Apr 4 '13 at 4:53
Isn't $|det(k)|=1$ for all $k \in K$? –  Marc Palm Apr 4 '13 at 10:58
Palm, yes. It is $K$ invariant function. –  James Apr 4 '13 at 12:01