TITLE: Two conditions for continuity?
QUESTION [1 upvotes]: Below, $f$ is a function $X\rightarrow X'$ between topological spaces. Are both those conditions equivalent with continuity?

$x\in (f^{-1}(A'))^i\Rightarrow f(x)\in (A^{\prime})^i$, for any $A'\subseteq X'$, where $B^i$ stands for the interior of B.
If $x$ is a cluster point of $(x_k)_\mathbb N$, then $f(x)$ is a cluster point of $(f(x_k))_\mathbb N$

REPLY [2 votes]: HINT: Consider what happens when $X$ is $\Bbb R$ with the cocountable topology, $X'$ is $\Bbb R$ with the discrete topology, and $f(x)=x$ for all $x\in\Bbb R$.


TITLE: A sum of a function and its derivatives at a point
QUESTION [4 upvotes]: Let f be a differentiable function on $\mathbb R$ satisfying $$f(x)=-(x^2-x+1)e^2+\int_0^x e^{x-y}f’(y)dy$$
If $f(1)+f’(1)+f''(1)=ke$, find $k$.

What I did: Call $\int_0^x e^{-y}f’(y)dy $ as $u$. Call $\int_0^1e^{-y}f’(y)dy$ as $u_1$.
$$f(x)=-(x^2-x+1)e^2+\int_0^x e^{x-y}f'(y)dy$$
$f(1)=-e^2+eu_1$,
$$f’(x)=-(2x-1)e^2+e^x(u+e^{-x}f’(x))$$
So $(2x-1)e^2=e^xu$. At x=1, $e^2=eu_1$, so $f’(1)=0$. (Is this useless?)
$$f''(x)=-2e^2+e^x(u+e^{-x}f'(x)+e^{-x}f''(x))$$
So $f'(x)=-e^xu+2e^2$ so $f'(1)=2e^2-eu_1$
So far, $f(1)+f'(1)=e^2$  How to find $f'(1)$? Is there anything which directly gives the sum $f(1)+f'(1)+f''(1)$?

REPLY [3 votes]: One has $$\displaystyle{f(x)=-(x^2-x+1)e^2+e^x\int_0^x e^{-y}f’(y)dy} \quad \quad \quad (*)$$ By direct computation, you get
\begin{align*} f'(x)& =-(2x-1)e^2 + e^x \int_0^x e^{-y}f’(y)dy + e^x e^{-x}f'(x) \\ & = -(2x-1)e^2 + e^x \int_0^x e^{-y}f’(y)dy + f'(x) \end{align*}
which simplifies and gives that $\displaystyle{ -(2x-1)e^2 + e^x \int_0^x e^{-y}f’(y)dy = 0}$, i.e. that $$e^x\int_0^x e^{-y}f’(y)dy =(2x-1)e^{2}$$
With this information, $(*)$ becomes $$f(x)= -(x^2-x+1)e^2+(2x-1)e^2 = e^2(-x^2+3x-2)$$
and you get successively $f'(x)=e^2(-2x+3)$ and $f''(x)=-2e^2$, so
$$f(1)+f'(1)+f''(1)=0+e^2-2e^2 = -e^2$$
so $$\boxed{f(1)+f'(1)+f''(1)=ke \quad \text{where} \quad k=-e}$$


TITLE: How find the value of $f(n)=P_{n-1}^{1}-P_{n-1}^{2}+P_{n-1}^3+\cdots+(-1)^{n}P_{n-1}^{n-1}$
QUESTION [0 upvotes]: let
$ $
show that $f(n)$  is Polynomial and such The coefficient of
is integer ,and Find the $f(n)$ close form.
where $$P_{n}^{r}=\dfrac{n!}{(n-r)!}$$
My try:
it is easy to know $P_{n}^{r}=(n)(n-1)(n-2)\cdots (n-r+1)$ is Polynomial of $n$,and the coefficient is integer,
so
I think $f(n)$ is Polynomial of $n$,and  the coefficient is integer too.But How find the close form $f(n)$?

REPLY [0 votes]: What this problem reduces to [If $n$ is even] is:
$$f(n)=(n-1)!{({1\over0!}-{1\over1!}+{1\over2!}-{1\over3!}+\cdots)}$$
Which becomes:
$$f(n)=(n-1)!{(e^{-1})}$$
This is not the exact answer but should give an approximate idea of the answer.


text:
\chapter{Depth}

In this chapter we shall study representations of $p$-adic groups, in particular properties of their depth. A powerful tool for that is
provided by Bruhat-Tits theory. We first briefly review some aspects of it in Section \ref{depth-backgr} and
then, in Section \ref{depth-main}, we prove the results we need for counting self-dual representation.
In Section \ref{depth-descent} we use the local descent to show a partial result towards depth preservation for split classical groups. However, depth need not be preserved in the case of wildly ramified groups, as we note in Section \ref{depth non}.
The last Section, \ref{depth restr}, is devoted to the study of the restriction of certain representations of $SO_{2n}$ to $SO_{2n-1}$, which is one of the steps needed in trying to extend the results of Section \ref{depth-descent} to a proof of full depth preservation, not just one inequality.

Throughout this chapter, all groups and representations are $p$-adic, unless stated otherwise.

\section{Background}\label{depth-backgr}

Throughout this chapter, let $F$ be a local non-archimedean field (of characteristic zero) and $G$ a connected reductive group over $F$.
We shall study irreducible admissible representations of $G(F)$, their depth and behaviour under functoriality, so let's first review
some background on Bruhat-Tits theory, Moy-Prasad filtrations, and depth. The basic references that we shall follow are \cite{tits}, \cite{yuottawa},
and \cite{mp2}.

Let $S$ be a maximal $F$-split torus of $G$, $N(S)$ the normalizer of $S$ in $G$, and $Z(S)$ the centralizer of $S$ in $G$.
Let $C$ be the maximal $F$-split torus contained in the center of $G$, we have $C\subset S$.
Let $(\Phi, R, \Phi^\vee, R^\vee)$ be the relative root system, let $W$ be its Weyl group and
$\widetilde{W}$ the affine Weyl group, an extension of $W$ by a free abelian group of rank $\dim S$.
The (restricted) apartment $A=A(G, S, F)$ is defined as an affine space under $(X_*(S)\otimes \mathbb R)/(X_*(C)\otimes \mathbb R)$, it is equipped by an action of $\widetilde W$
and by a corresponding action of $N(S)$. We denote both of these action by $\nu$. See \cite {tits}, 1.2 for details.
Note that \cite {tits} works with the extended apartment, but the reduced one is somewhat more convenient for our purposes.

We also have the set of affine roots $\Phi_{\mathrm{af}}$, being the set of certain affine functions $\alpha=a+\gamma$ for $a\in\Phi$ and $\gamma\in\mathbb R$.
See \cite {tits}, 1.6.

For $a\in\Phi$ we have the root subgroup $U_a$; for an affine function $\alpha=a+r$, $r\in\mathbb R$ one defines
a subgroup $X_\alpha$ of $U_a$ as the subset of elements $u$ satisfying $u=1$ or $\alpha(a, u)\geq \alpha$, where $\alpha(a, u)$ is a certain affine
function, defined in \cite {tits}, 1.4.

For an affine function $\alpha=a+r$ with $a\in\Phi$, we denote $A_\alpha=\alpha^{-1}([0, \infty))$,
its boundary is $\partial A_\alpha=\alpha^{-1}(0)$, and $r_\alpha$ is the affine reflection
whose vector part is the reflection $r_a$ and whose fixed hyperplane is $\partial A_\alpha$.
When $\alpha\in\Phi_{\mathrm{af}}$, we call $A_\alpha$ the half-apartments and
$\partial A_\alpha$ the walls. The chambers are the connected components of
the complement in $A$ of the walls (cf. \cite{tits}, 1.7).
A chamber is a polysimplex (i.e., a product of simplices); by a facet of $A$ we mean just
a facet of some chamber (viewed as a polysimplex).

The (restricted) building $\mathcal B=\mathcal B(G, F)$ is defined as the unique set equipped with a left $G$-action such that
$\mathcal B=\bigcup_{g\in G} gA$, the group $N(S)$ stabilizes $A$ and acts on it by $\nu$, and for every affine root $\alpha$, the group
$X_\alpha$ fixes the half-apartment $A_\alpha$ (cf. \cite{tits}, 2.1).

Fix a chamber $\mathcal C$. Then each element of $A$ is $N(S)$-conjugate to an element of the closure $\bar{\mathcal C}$. Hence every element of $\mathcal B$ is $G$-conjugate to an element of the closure $\bar{\mathcal C}$.

For a facet $\mathcal F$, one can define a corresponding parahoric subgroup $G_\mathcal F$. The definition is somewhat involved (\cite{yuottawa}, 2.2.4),
but roughly speaking, the parahoric subgroup is essentially the stabilizer of $\mathcal F$ in $G$.
One can also define the parahoric subgroup $G_x$ for $x\in\mathcal B$;
if $x$ is a ``generic" point of a facet $\mathcal F$, then $G_\mathcal F=G_x$.

\

We have the Moy-Prasad subgroups which are used to define the depth of admissible representations \cite{mp2}.
First, denote by $Z_n$ the group of all $z\in Z(S)$ such that $\omega(\chi(z)-1)\geq n$ for all characters $\chi$ of $Z(S)$
($\omega$ is the additive valuation on $F$).
For $x\in \mathcal B$ and $r\geq 0$, we define
$G_{x, r}$ to be the intersection of $G_x$ with the subgroup generated by $Z_n$ for $n\geq r$ and by
$X_\alpha$ for all $\alpha$ satisfying $\alpha(x)\geq r$.
Denote by $G_{x, r+}$ the union of all $G_{x, s}$ with $s>r$.

Let $\pi$ be an irreducible admissible representation of $G$.
The depth of $\pi$ is defined as
$\depth\pi=\inf\{r\geq 0| \exists x\in\mathcal B$ such that $\pi^{G_{x, r+}}\neq 0\}$.

One also defines the depth of a Weil-Deligne
representation $\varphi: W_F^\prime\rightarrow$ $^LG$ as $\inf\{r\ |\ \varphi(I_s)=1$ for all $s>r \}$, where $I_s$, $s\in\mathbb Q$, is the filtration
of the inertia subgroup $I$ of $W_F\subset W_F^\prime$.

We have the following fundamental properties of depth:
\begin{itemize}
\item $\depth\pi$ is a rational number and the infimum is achieved for some $x\in\mathcal B$.
\item The depth is preserved by parabolic induction and the Jacquet functor. Namely, let
$M$ be a Levi of $G$ and $\sigma$ an irreducible admissible representation of $M$.
If $\pi$ is an irreducible subquotient of the normalized induction $\mathrm{Ind}_M^G\sigma$, then $\depth\sigma=\depth\pi$
(and similarly for the Jacquet functor).
\item Depth preservation: Assume that $\mathbf G$ is unramified or tamely ramified. If $\varphi:W_F^\prime\rightarrow$ $^LG$ is the Langlands parameter of a representation $\pi$, then it is conjectured that $\depth\pi=\depth\phi$. J.-K. Yu \cite{yuottawa} has proved this
for $GL_N$ and for tamely ramified tori.
\end{itemize}

As we note in Section \ref{depth non}, depth need not be preserved in the case of wildly ramified groups. This has also been observed by Gross-Reeder \cite{gross-reeder};
Reeder-Yu \cite{reeder-yu} recently provided more examples of this phenomenon.

\section{Bounds on Depth}\label{depth-main}

We will use the following result on uniform admissibility, due originally to Howe when $\mathbf G=GL_N$, and
then to Bernstein for general $\mathbf G$ (with a simpler proof) -- for part b) see \cite{bernstein}, Theorem 1; part a) then follows as in \cite{bz}, Corollary 4.15.

\begin{theorem}\label{Bernstein}(\cite{bernstein}, \cite{bz})
Let $\mathbf G$ be a reductive group over a non-archimedean local field $F$. Let $K$ be an open compact subgroup of $G=\mathbf G(F)$. Then:

a) There are finitely many supercuspidal representations $\sigma$ of $G$ such that $\sigma^K\neq 0$.

b) There is an $n\in\mathbb N$ (depending only on $G$ and $K$) such that, for all
irreducible admissible representations $\pi$ of $G$, we have $\dim \pi^K\leq n$.
\end{theorem}

Both parts of Theorem \ref{Bernstein} will be useful for us in the thesis.
We shall shortly use the first part to prove part b) of Proposition \ref{bound-depth}.
The second part of the theorem will play an important role in the arguments of Chapter 5, where it will enable us to uniformly bound the dimensions of the spaces of $K$-fixed vectors of the self-dual representations which we are counting.

\begin{proposition}\label{bound-depth}
Let $\mathbf G$ be a reductive group over a non-archimedean local field $F$.

a) For each $r\geq 0$ there is an open compact subgroup $K$ of $G=\mathbf G(F)$
such that if $\depth\pi\leq r$, then $\pi^K\neq 0$.
When $\mathbf G$ is split and almost simple, we can take $K$ to be the principal congruence subgroup $K_m$
with $m=\left\lceil r \right\rceil +1$.

b) There are finitely many supercuspidal representations $\sigma$ of $G$ such that  $\depth\sigma$  $\leq r$.
\end{proposition}

\begin{proof}
a) In the split case this is due to Radhika Ganapathy \cite{ganapathy}, Lemma 8.2. The proof in the general case is similar to the split one:

Let $\sigma$ be an irreducible admissible representation of $G$ with $\depth\sigma\leq r$.

Let $\mathcal C$ be a chamber of $\mathcal B$.
For $a\in\Phi$,  $a(x)$ is a continuous function on the compact set $\bar{\mathcal C}$, and so it has a minimum,
which we denote $m(a)$. Define then $G_r$ to be the subgroup of $G$ generated by all the
$Z_n$ with $n>r$ and by all $U_\alpha$ with $\alpha=a+\gamma$
satisfying $\gamma>r-m(a)$. We shall show that $\sigma$ has a vector fixed by $K=G_r$.

For each $x\in\mathcal B$ there is $y\in\bar{\mathcal C}$ and $g\in G$ such that $x=g\cdot y$. Then $G_{x, r}=gG_{y, r}g^{-1}$. Thus if $\depth\sigma\leq r$, there is $x\in \mathcal B$ such that $\sigma^{G_{x, r+}}\neq 0$, and therefore also
$\sigma^{G_{y, r+}}\neq 0$.

Take now $\alpha=a+\gamma$ such that $\gamma> r-m(a)$. We have $\alpha(y)=a(y)+\gamma> m(a)+(r-m(a))=r$, and so we see that the subgroup
$U_\alpha$ is contained in $G_{y, r+}$. Since this is true for any such $\alpha$ and also for all $Z_n$ with $n>r$, we conclude that
$G_r\subset G_{y, r+}$. Hence $\sigma^{G_{r}}\neq 0$.

b) Take the subgroup $K=G_r$ from part a). By Theorem \ref{Bernstein} a) there are only finitely many supercuspidals having a vector fixed by $K$,
and so there are also only finitely many supercuspidals of depth $\leq r$.
\end{proof}

Let $\mathbf G$ be a quasisplit classical group over a non-archimedean local field $F$, i.e., $G$ is split $SO_{2n+1}$, $Sp_{2n}$ or $SO_{2n}$, or quasisplit (non-split) even orthogonal corresponding to a quadratic extension $E/F$. By Arthur's work we know the local Langlands correspondence for $G$ (conditionally on the stabilization of the twisted trace formula) and a slightly weaker form of it when $\mathbf G$ is even orthogonal. We know that irreducible admissible representations of $G$ lift to self-dual (irreducible admissible) representations of $GL_N(F)$ for a suitable $N$.

Let $^L G$ be the Langlands dual group of $G$ and $W_F^\prime$ the Weil-Deligne group. For a representation $\pi$ of $G$ we denote $\phi_\pi: W_F^\prime \rightarrow$ $^L G$ the corresponding Langlands parameter and $\Pi$ the lift to $GL_N(F)$ with parameter $\iota\circ\phi_\pi: W_F^\prime \rightarrow$ $^L G\rightarrow$ $^L GL_N$ (all this being up to outer conjugacy by $O_{2n}$ in the even orthogonal case).
Note that $\depth\phi_\pi=\depth\iota\circ\phi_\pi$.

\begin{proposition}\label{rs}
Fix $\mathbf G$ and $F$ as above. Then for each $r\geq 0$ there exists $s\geq 0$ such that:

a) For all irreducible admissible representations $\pi$ of $G$  if $\depth \phi_\pi=\depth \Pi\leq r$, then $\depth\pi\leq s$.

b)  For all irreducible admissible representations $\pi$ of $G$ if $\depth\pi\leq r$, then $\depth \phi_\pi=\depth \Pi\leq s$.
\end{proposition}

\begin{proof}
a) Assume first that $G$ is not an even orthogonal group, so that we know the full local Langlands correspondence for $G$.

1) We first prove the statement for supercuspidal representations. Let $\sigma$ be a supercuspidal representation of $G$. Let $\phi_\sigma$ be its Langlands parameter and $\sigma^\prime$ the lift to $GL_N(F)$. Assume that $\depth \phi_\sigma=\depth \sigma^\prime\leq r$.

The representation $\sigma^\prime$ is not necessarily supercuspidal, but there will be a Levi $M$ of $GL_N(F)$ and a supercuspidal representation $\tau$ of $M$ so that $\sigma^\prime$ is a subquotient of the normalized parabolic induction $\mathrm{Ind}_M^{GL_N(F)} \tau$. By the general properties of depth, we have
$\depth \tau=\depth \sigma^\prime\leq r$.

Now, there are only finitely many standard Levis $M$ (including the one $M=GL_N(F)$).
By Proposition \ref{bound-depth}, for a fixed $M$ there are only finitely many supercuspidal representations $\tau$ of $M$ with $\depth\tau\leq r$.
And for fixed $M$ and $\tau$, the parabolic induction $\mathrm{Ind}_M^{GL_N(F)} \tau$ has only finitely many possible constituents $\sigma^\prime$ by \cite{cass}, Corollary 6.3.7.
Let $\Sigma^\prime$ be the finite set of all triples $(M, \tau, \sigma^\prime)$ as above, and let $\Sigma$ be the finite set of
supercuspidal representations of $G(F)$ that lift to some $\sigma^\prime$ belonging to a triple $(M, \tau, \sigma^\prime)\in\Sigma^\prime$. Define $s$ as the maximum of the depths of all representations $\sigma_0\in\Sigma$.

Thus the representation $\sigma$ with which we started lies in $\Sigma$, and so $\depth\sigma\leq s$.

2) Now let $\sigma$ be a supercuspidal representation of some (standard) Levi subgroup $M$ of $G(F)$.

Then $M=\mathbf G_{k_0}(F)\times GL_{k_1}(F)\times\cdots\times GL_{k_l}(F)$, where $\mathbf G_{k_0}$ is a group of the same type as $\mathbf G$ and smaller rank $k_0$. Correspondingly we have $\sigma=\sigma_0\times\sigma_1\times\cdots\times\sigma_l$, where $\sigma_i$ are supercuspidal representations of $G_{k_0}$ or $GL_{k_i}(F)$. Let $\phi_i$ be the Langlands parameter of $\sigma_i$, $\sigma_0^\prime$ the lift of $\sigma_0$ to suitable $GL_{k_0^\prime}(F)$, and $M^\prime=GL_{k_0^\prime}(F)\times GL_{k_1}(F)\times\cdots\times GL_{k_l}(F)$. Then $\phi=\phi_0\oplus\phi_1\oplus\cdots\oplus\phi_l$ is the Langlands parameter of $\sigma$.

We have $\depth\sigma=\max(\depth\sigma_i)$ and $\depth\phi=\max(\depth\phi_i)$. By the depth preservation theorem for $GL$, we know that $\depth\sigma_i=\depth\phi_i$ for $i>0$. By the first part of the proof we now get the statement of the proposition in this case.

Since there are finitely many standard Levi subgroups, we can define $s$ to be the maximum of numbers $s_M$ obtained above for individual Levis $M$.

3) Finally, let $\pi$ be any representation of $G(F)$ with $\depth \phi_\pi=\depth \Pi\leq r$.
There is a standard Levi $M$ of $G$ and a supercuspidal representation $\sigma$ of $M$ such that $\pi$ is a subquotient of the normalized induction $\mathrm{Ind}_M^G \sigma$. By the properties of depth we have $\depth \sigma=\depth\pi$. Let $\phi_\sigma$ be the Langlands parameter of $\sigma$. We can also see that
$\depth \phi_\sigma=\depth\phi_\pi$. By our assumption, this depth $\leq r$, and so $\depth \sigma=\depth\pi\leq s$, as we wanted to show.

4) Now it remains to deal with the even orthogonal case. In this case we know the local Langlands correspondence only up to outer conjugacy by $O_{2n}$.
Given a representation $\pi$ of $G$, we know that its Langlands parameter will be one of two possible parameters $\phi_1$, $\phi_2$. These parameters are conjugate by $O_{2n}$ in $^L G$, and so $\depth\phi_1=\depth\phi_2$. Now we can use the proof above both for $\phi_1$ and $\phi_2$ and define $s$ to be the maximum of the two numbers $s_1, s_2$ that we obtain.

b) The proof is entirely analogous to part a), so we shall omit it.

\end{proof}

For $m\in\mathbb N$ denote by $K_m$ the principal congruence subgroup of $GL_N(F)$ of level $m$, i.e., $K_m$ consists of matrices congruent to the
identity $I$ modulo $\wp^m$ (where $\wp$ is the prime ideal of $\mathcal O_F$).

\begin{corollary}\label{depth cor}
Let $\mathbf G$ be a quasisplit classical group.
For each $m\in\mathbb N$ there exists a compact open subgroup $K\subset G$ such that:

For each self-dual irreducible admissible representation $\Pi$ of $GL_N(F)$ which descends to a representation $\pi$ of $G$
and satisfies $\Pi^{K_m}\neq 0$, one has $\pi^K\neq 0$.

\end{corollary}

\begin{proof}
Since $\Pi^{K_m}\neq 0$, we see that $\depth\Pi\leq m$. Thus by Proposition \ref{rs}, there is $s$ (depending only on $m$ and not on $\Pi$)
such that $\depth\pi\leq s$. By Proposition \ref{bound-depth} a), there is $K$ (again depending only on $s$) such that $\pi^K\neq 0$.
\end{proof}

\section{Local Descent}\label{depth-descent}

We can also use the construction of local descent to obtain a sharper estimate on the depths in Proposition \ref{rs}
in the case of generic representations.

\begin{theorem}\label{descent-bound}
Let $F$ be a $p$-adic field and $G=SO_{2n+1}(F)$.
Assume that $p$ is sufficiently large.

a) Let $\Pi$ be an (irreducible admissible) representation of $GL_n(F)$ that descends to a (generic) representation $\pi$ of $G$. Then $\depth\Pi\geq\depth\pi$.

b) Let $\pi$ be an (irreducible admissible) generic representation of $G$ and $\phi_\pi$ its Langlands parameter. Then $\depth\pi\leq\depth\phi_\pi$.
\end{theorem}

Before proving the theorem,
let us review the local descent for the case $\mathbf G=SO_{2n+1}$, following \cite{js}.

Let $\tau$ be a self-dual supercuspidal representation of $GL_{2n}(F)$  that descends to $G=SO_{2n+1}(F)$ (rather than to an even orthogonal group). Denote $H=SO_{4n}(F)$.
First view $GL_{2n}(F)$ as the Siegel Levi of a parabolic $P$ in $H$ and take the normalized induction $\rho_\tau=\mathrm{Ind}_P^H \tau |\det|^{1/2}$. Let $\pi_\tau$ be the Langlands quotient of $\rho_\tau$.

The group $H=SO_{4n}(F)$ is the group of isometries of a $4n$-dimensional vector space $V$ preserving
a symmetric bilinear form $b$. Fix maximal isometric subspaces $V^+$ and $V^-$ of $V$ in duality with respect to $b$. Also fix a maximal flag
$0\subset V_1^+\subset V_2^+\subset\dots\subset V_n^+=V^+$ and a corresponding basis
$\{e_1, \dots, e_n\}$ such that $V_i^+=\mathrm{Span} (e_1, \dots, e_i)$. Let $\{e_{-1}, \dots, e_{-n}\}$ be the dual basis of $V^-$ and denote $V_{i}^-=\mathrm{Span} (e_{-1}, \dots, e_{-i})$.
Finally, let $W=(V_{n-1}^+ + V_{n-1}^-)^\perp$.

Let $P_{n-1}=M_{n-1}N_{n-1}$ be the parabolic subgroup of $H$ stabilizing the flag
$0\subset V_1^+\subset V_2^+\subset\dots\subset V_{n-1}^+$. Then $M_{n-1}$ is isomorphic
to $SO(W)\times GL_1(F)^{n-1}$, note that $SO(W)\simeq SO_{2n+2}(F)$.

Fix $\alpha\in F^\times$ and denote $y=e_{2n}-\alpha^2 e_{-2n}$. Let $L_{n-1}$ be
the stabilizer of $y$ in $SO(W)$, we have
$L_{n-1}=SO(\{y\}^\perp\cap W)\simeq SO_{2n+1}(F)(\subset SO_{2n+2}(F))$
(for more detail on this embedding, see Section \ref{depth restr}).

Let $\psi$ be an unramified additive character of $F$ and let $\psi_{n-1}$ be the corresponding
character of $N_{n-1}$. For a smooth representation $\pi$ of $H$
we shall denote by $J_{\psi_{n-1}}(\pi)$ the twisted Jacquet module of $\pi$ with respect to $N_{n-1}(F)$ and its character $\psi_{n-1}$, i.e.,
$J_{\psi_{n-1}}(\pi)=V_\pi / \mathrm{Span}\{\pi(a)v-\psi_{n-1}(a)v | a\in N_{n-1}, v\in V_\pi\}$.
Note that, as a representation of the Levi $M_{n-1}$, this is the same as the (usual, non-twisted) Jacquet module
$J(\pi\otimes \psi_{n-1}^{-1})=V_\pi / \mathrm{Span}\{\pi(a)\psi_{n-1}(a)^{-1}v-v | a\in N_{n-1}, v\in V_\pi\}$.

Finally, let $\sigma=\sigma_{\psi, n-1}(\tau)$ be the restriction of the twisted Jacquet module to $L_{n-1}$, $\sigma=\mathrm{Res}_{L_{n-1}}J_{\psi_{n-1}}(\pi_{\tau})$.

Then $\sigma$ is the descent of $\tau$ and it is an irreducible generic supercuspidal representation of $L_{n-1}\simeq SO_{2n+1}(F)$.

Let us now prove Theorem \ref{descent-bound}; note that the local descent construction applies in the cases of other classical groups as well,
only generally it may happen that $\sigma$ is not irreducible, and the descent is in fact an irreducible subquotient of $\sigma$.
The result and its proof are similar, so we deal only with the case of $G=SO_{2n+1}$ for simplicity of notation.

\begin{proof}
a) It suffices to prove the theorem only for supercuspidal representations, in general one proceeds as in the proof of Proposition \ref{rs}, part 3).

So let us take a supercuspidal self-dual representation $\tau$ of $GL_{2n}(F)$ that descends to $G=SO_{2n+1}(F)$.
The character $|\det|^{1/2}$ is trivial on $GL_{2n}(\mathcal O)$, and so it has depth zero, and hence $\depth\tau=\depth\tau |\det|^{1/2}$.
Since parabolic induction preserves depth and $\pi_\tau$ is an irreducible subquotient of $\rho_\tau=\mathrm{Ind}_P^H \tau |\det|^{1/2}$,
we see that $\depth\tau=\depth\pi_\tau$.

The character $\psi_{n-1}^{-1}$ is unramified, and so
$\depth\pi_\tau=\depth(\pi_\tau\otimes \psi_{n-1}^{-1})$. Since depth is preserved under
taking a Jacquet quotient, we see that
$\depth\tau=\depth(\pi_\tau\otimes \psi_{n-1}^{-1})=\depth (J_{\psi_{n-1}}(\pi_{\tau}))$,
where we consider $J=J_{\psi_{n-1}}(\pi_{\tau})$ as a representation of $M_{n-1}$.

We are assuming that the residual characteristic $p$ is large enough, and so the
Moy-Prasad filtration on $L_{n-1}\simeq SO_{2n+1}(F)$ is obtained as the intersection of
the filtration on $M_{n-1}\simeq SO_{2n+2}(F)$ with $L_{n-1}$ (see \cite{yuottawa}, 2.3.10.3).
Clearly, if $J$ has a vector fixed by some subgroup $(M_{n-1})_{x,r}$, then it has also a vector
fixed by $(M_{n-1})_{x,r}\cap L_{n-1}$. Thus $\depth \mathrm{Res}_{L_{n-1}}J\leq \depth J=\depth\tau$,
as we wanted to prove.

b) Easily follows from a).
\end{proof}

Let us note that the reason for why we get only one inequality towards depth preservation is the
last step of taking the restriction, which
may generally decrease the depth. It seems plausible that studying the local descent more closely can prove
the full depth preservation for generic representations -- we outline a partial progress in this direction in Section \ref{depth restr}.

\section{Depth Preservation and Wildly Ramified Groups}\label{depth non}

In this section we give an example when depth is not preserved under
Langlands reciprocity and functoriality.
This occurs in some of the examples of residual characteristic $p=2$ discussed in Section 7.2 of \cite{cps}.
The fact that depth need not be preserved when residual characteristic is small has already been observed by Gross-Reeder \cite{gross-reeder};
Reeder-Yu \cite{reeder-yu} recently provided more examples of this phenomenon.

Let $F=\mathbb Q_p$ and $E=\mathbb Q_p(\sqrt {p})$. Let $\mathbf G$ be the non-split, quasisplit orthogonal group $SO_{2n}^*$ attached to the quadratic extension
$E/F$, let $G=\mathbf G(F)$.

Let $\eta=\eta_{E/F}: F^\times \rightarrow \mathbb C$ be the quadratic character associated to $E/F$ by local class field theory.
Note that $\eta$ is defined by $\ker \eta=N_{E/F}(E^\times)$.

It is easy (and classical) to observe that $\eta$ is ramified and to compute its depth:

\begin{lemma}\label{eta}
$\eta$ is ramified. If $p>2$, then $\depth\eta=0$. If $p=2$, then $\depth\eta=2$.
\end{lemma}

\begin{proof}
We want to check that $\eta|\mathbb Z_p^\times\neq 1$. Take $a\in\mathbb Z_p^\times$ such that $a$ is not a square modulo $p$.
We shall show that $a$ is not a norm. Assume false, i.e., we have $a=N(x+\sqrt {p} y)=x^2-py^2$ for some $x, y\in\mathbb Q_p$. Let $p^m$ be the common denominator of $x$ and $y$ and write
$x=p^{-m}b, y=p^{-m}c$ with $b, c\in\mathbb Z_p$ and $m\geq 0$. We obtain $p^{2m}a=b^2-pc^2$.

If $m>0$, then $p$ divides $b$ and we can write $b=pd$ and we get $p^{2m-1}a=pd^2-c^2$. Thus $c=pe$ and
$p^{2m-2}a=d^2-pe^2$. Proceeding in this way, we eventually obtain an equation with $m=0$.

Let's thus assume that $m=0$ and $a=b^2-pc^2$. We see that if $a$ is a norm, it has to be a square modulo $p$.

Hence $\eta$ is non-trivial on $\mathbb Z_p^\times$, i.e., $\eta$ is ramified.

The level $r$ subgroup $G_r$ of the Moy-Prasad filtration on $\mathbb Q_p^\times$ is just
$1+p^{\lceil r\rceil}\mathbb Z_p$, where $\lceil r\rceil$ is the smallest integer with $\lceil r\rceil\geq r$.
We then have $G_{r+}=\bigcup_{s>r}G_s=1+p^{r+1}\mathbb Z_p$ for $r\in\mathbb N$, and
$G_{r+}=G_r=1+p^{\lceil r\rceil}\mathbb Z_p$ for $r\not\in\mathbb N$.

When $p>2$, we can see by Hensel's Lemma that each element of $1+p\mathbb Z_p$ is a norm, and so $\depth\eta=0$.
When $p=2$, every element of $1+8\mathbb Z_2$ is a norm, whereas 5 is not a norm. Hence
$\depth\eta=2$.
\end{proof}

Let now $p=2$.
Take any irreducible, admissible and generic representation $\pi$ of $G$ of depth zero and let $\Pi$ be its lift to $GL_{2n}(\mathbb Q_2)$.
The central character of $\Pi$ is $\eta$, and so $\depth\Pi\geq\depth\eta=2$.
In fact, in the explicit cases discussed in Section 7.2 of \cite{cps}, we see that  $\depth\Pi=2$ (take for example $\mu_1=\dots=\mu_n=1$, then $\Pi$ is just the induction
of the depth one character $(1, \dots, 1, \eta, 1, \dots, 1)$ with $\eta$ at the $n$-th coordinate,
of the torus $GL_1^{2n}(\mathbb Q_2)$).

Note that $\Pi$ is not unramified, even when $\pi$ was. This is no contradiction,
as the fact that unramified representations lift to unramified ones (via Satake isomorphism) holds only for quasisplit groups which split over an unramified extension, i.e., an unramified group, which is not our case. To sum it up, we have:

\begin{corollary}\label{depth non-pres}
Let $\pi$ be an irreducible, admissible, generic, unramified representation of the ramified group $G=SO_{2n}^*(\mathbb Q_2)$ and $\Pi$ its functorial lift to $GL_{2n}(\mathbb Q_2)$.
Then $\Pi$ is not unramified and $\depth\pi=0<2\leq\depth\Pi$.
\end{corollary}

\section{Restriction and Depth}\label{depth restr}

In the local descent construction in the previous section, we obtained a representation $J_{\psi_{n-1,\alpha}}(\pi_{\tau})$ of
$M_{n-1}\simeq SO_{2n+2}(F)\times (GL_1(F))^{n-1}$ and restricted it to $SO_{2n+1}(F)\subset SO_{2n+2}(F)$. The depth of the representation may have
decreased in this step. The representation of $M_{n-1}$ is a product
$\sigma\otimes \chi_1\otimes\cdots\otimes \chi_{n-1}$, where $\sigma$ is a representation of $SO_{2n+2}(F)$ and $\chi_i$ are characters of $GL_1(F)$.

Hence if we wanted to show that depth does not decrease under this restriction,
first we would need to know what are the characters $\chi_i$, e.g., whether they are maybe always unramified in the descent to $SO_{2n+1}$. The author doesn't know this at present.

Second, we need to understand the restriction of $\sigma$ from $SO_{2n+2}(F)$ to $SO_{2n+1}(F)$. In this section we show that the depth
indeed does not change under this restriction. Let us discuss now the situation slightly more generally:

\

In the rest of this section let $F$ be a $p$-adic field, $p\neq 2$. We will consider the (split) orthogonal group of a quadratic space $V$.
Let $V$ be an $N$-dimensional $F$-vector space. If $N=2n$, choose a basis $x_1, \dots, x_{n}, x_{-1}, \dots, x_{-n}$ of $V$ and equip
$V$ with the quadratic form $Q(\sum_{i=-n}^{n} a_ix_i)=\sum_{i=1}^n a_i a_{-i}$.
If $N=2n+1$, choose a basis $x_0, x_1, \dots, x_{n}, x_{-1}, \dots, x_{-n}$ of $V$ and equip $V$ with the quadratic form $Q(\sum_{i=-n}^{n} a_ix_i)$ $=a_0^2+\sum_{i=1}^n a_i a_{-i}$.
In both cases $SO(V)=\{g\in SL(V) | gv=v$ for all $v\in V\}=SO_N(F)$.

We'll be considering the embeddings $SO_{2n-1}(F)\subset SO_{2n}(F)\subset SO_{2n+1}(F)$ defined as follows: Take a vector $v\in V$ such that $Q(v)\neq 0$ and take $W=\{v\}^\perp$. Then $SO(W)=SO_{N-1}(F)$ is the stabilizer of $v$ in $SO(V)$.

\

From now on assume that $N=2n$ is even and take $v=x_n-a^2x_{-n}$ for some $a\in F^\times$ (this is the situation that
occurs in the local descent from $GL_{2n-2}$ to $SO_{2n-1}$).

\begin{proposition}
Let $SO(W)=SO_{2n-1}(F)\subset SO(V)=SO_{2n}(F)$ be an embedding as above. Then there is a quadratic space $U\supset V$ with a quadratic form extending
the one on $V$ (hence we will also denote it by $Q$) such that $SO(W)\times GL_1(F)$ is a Levi subgroup of $SO(U)=SO_{2n+1}(F)$
(where we take $SO(W)\subset SO(U)$ by the natural inclusion). Moreover, $GL_1(F)\cap SO(V)=\{1\}$.
\end{proposition}

\begin{proof}
Take $U$ to be the vector space with basis $x_0, x_1, \dots, x_{n}, x_{-1}, \dots, x_{-n}$
(where $x_1, \dots, x_{n}, x_{-1}, \dots, x_{-n}$ is a basis of $V$ and $x_0\not\in V$),
extend the quadratic form to $U$ by $Q(\sum_{i=0}^{2n} a_ix_i)=a_0^2+\sum_{i=1}^n a_i a_{n+i}$.
Then $V$ is the orthogonal complement of $x_0$ in $U$, which gives $SO(V)=SO_{2n}(F)\subset SO(U)=SO_{2n+1}(F)$.

To show that $SO(W)\times GL_1(F)$ is a Levi in $SO(U)=SO_{2n+1}(F)$, we want to find a corresponding isotropic flag $0\subset U_0\subset U$
fixed by a parabolic $P$.
Let $U_0$ be the 1-dimensional subspace spanned by $u_0=ax_0+x_n-a^2x_{-n}$. We have $Q(u_0)=a^2-a^2=0$, and so $U_0$ is isotropic
and $SO(U_0)=GL_1(F)$. By definition, $SO(W)$ fixes $x_0$ and $x_n-a^2x_{-n}$, and so $SO(W)\times GL_1(F)\subset P$ is the Levi.
\end{proof}

\begin{proposition}\label{restr compact}
Let $\pi$ be an irreducible admissible representation of $G=SO_{2n}(F)$ $=SO(V)$ such that its restriction
$\pi^\prime$ to $H=SO(W)=SO_{2n-1}(F)\subset G$ is also irreducible (and automatically admissible). Assume moreover that $\pi^\prime$ is supercuspidal and generic.

a) Let $\chi$ be a unitary character of $F^\times$ such that $\chi^2\neq \mathbf 1$. Then $\mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)} \pi^\prime\otimes \chi\simeq \pi\otimes \chi$.

b) Let $K$ be a compact open subgroup of $G$. Then $\pi^K\neq \{0\}$ if and only if $\pi^{\prime K\cap H}\neq \{0\}$.
\end{proposition}

\begin{proof}
a) Take $U$ as in the previous proposition so that $H\times GL_1(F)$ is a Levi subgroup of $SO(U)=SO_{2n+1}(F)$
, where we take $H=SO(W)\subset G=SO(V)\subset SO(U)$ by the natural inclusions.
Consider the parabolic induction $\mathrm{Ind}_{H\times GL_1(F)}^{SO(U)} \pi^\prime$ $\otimes \chi$. By Lemma \ref{lemma induction},
this induced representation is irreducible.

Now $$\mathrm{Ind}_{H\times GL_1(F)}^{SO(U)} \pi^\prime \otimes \chi=\mathrm{Ind}_{G\times GL_1(F)}^{SO(U)} \mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)}\pi^\prime \otimes \chi$$
is irreducible, and so
$\mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)}\pi^\prime \otimes \chi$ is also irreducible.

By Frobenius reciprocity we get that $$\mathrm{Hom}_{G\times GL_1(F)}(\pi\otimes \chi, \mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)}\pi^\prime \otimes \chi)\simeq \mathrm{Hom}_{H\times GL_1(F)}(\pi^\prime \otimes \chi, \pi^\prime \otimes \chi),$$ because the restriction of $\pi \otimes \chi$ to $H\times GL_1(F)$ is $\pi^\prime \otimes \chi$. Since $\pi^\prime \otimes \chi$, $\pi\otimes \chi$, and $\mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)}\pi^\prime \otimes \chi$ are all irreducible, we see that
$\mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)} \pi^\prime\otimes \chi\simeq \pi\otimes \chi$.

b) If $\pi^K\neq\{0\}$, then take a non-zero vector $v$ fixed by $K$. Then clearly $v\in \pi^{\prime K\cap H}$.

Assume now that $\pi^{\prime K\cap H}\neq \{0\}$.
Let $K_0$ be an open compact subgroup of $F^\times$ such that $\chi(K_0)=1$.

Then using part a) we have
$$\{0\}\neq
\mathrm{Ind}_{(K\cap H)\times K_0}^{K\times K_0}\pi^{\prime K\cap H} \otimes \chi^{K_0}=
\left (\mathrm{Ind}_{H\times GL_1(F)}^{G\times GL_1(F)}\pi^{\prime} \otimes \chi\right )^{K\times K_0}=\left (\pi \otimes \chi\right )^{K\times K_0}.$$
Since $\chi(K_0)=1$, the space $\left(\pi \otimes \chi\right )^{K\times K_0}$ is equal to $\pi^K$,
and so $\pi^K$ is also non-zero.
\end{proof}

config: arxiv


TITLE: Renormalising $\Delta$ baryon mass in chiral effective field theory
QUESTION [2 upvotes]: I have essentially no experience of quantum field theory, other than a superficial knowledge of some basic ideas - my apologies if I've phrased anything unusually or made any mistakes in my question here.
For context, I've been looking at matrix methods in quantum mechanics, and I'm trying to understand a paper on a matrix model for $\Delta\to N\pi$ decay that uses chiral effective field theory to construct the Hamiltonian:
https://arxiv.org/abs/1303.4157
In the paper, the bare mass $\Delta_0$ of the $\Delta$ is adjusted to account for the $\Delta\to N\pi$ interaction. This is done by:
$$\Delta_0=E_\text{res}-\Sigma_{\Delta N}(k_\text{res})$$
where $E_\text{res}$ is the phenomenological value of the $\Delta$ resonance, $k_\text{res}$ is the corresponding pion momentum $E_\text{res}=\sqrt{k_\text{res}^2+m_\pi^2}$, and $\Sigma_{\Delta N}$ is given by the integral:
$$\Sigma_{\Delta N}(k)=\int_0^\infty dk'\frac{k'^2g_{\Delta N}^2(k')}{\omega_\pi(k')\left[\omega_\pi(k)-\omega_\pi(k')-i\epsilon\right]}$$
where $\omega_\pi(k)=\sqrt{k^2+m_\pi^2}$ is the pion energy and $g_{\Delta N}^2$ is the $\Delta- N\pi$ coupling (which I understand to be a sort of potential energy). It is said to be related to this diagram:

My question is essentially: how does one construct these integrals from these diagrams? For example, I am in particular interested in how to construct the integral for the self-energy associated with $\Delta\to\Delta \pi$ interaction, with the diagram:

REPLY [0 votes]: Although the paper mostly deals with Quenched QCD, Young et al. (https://arxiv.org/abs/hep-lat/0205017) does have some discussion of the Sigma Commutator in full QCD which is relevant here.
In the paper you referenced, they make use of a $\chi_{\Delta}$ in the $g_{\Delta N}^2$ term. Using the notation from Young et al., $\chi_{\Delta}$ from Hall et al. can be written as
$$\chi_{\Delta N} = \frac{3}{32\pi f_{\pi}^2}G_{\Delta N}\,,$$
where $G_{\Delta N} = \frac{2}{9}\mathcal C^2$ (Young uses $G_{\Delta N} = \frac{8}{25}(F+D)^2$, but that can be reduced to $\frac{2}{9}\mathcal C^2$ using the approximation $F = \frac{2}{3}D$).
To find the form of the Sigma Commutator for the $\Delta\to\Delta+\pi$ interaction then, you just have to use $\chi_{\Delta\Delta}$ instead, with $G_{\Delta\Delta} = (F+D)^2$. This then gives you an expression for $g_{\Delta\Delta}^2$, which can be used to find $\Sigma_{\Delta\Delta}(k)$.
For further discussion on the Sigma Commutator in a bit more general terms, you may want to try https://arxiv.org/abs/nucl-th/0005003.


TITLE: How many perfect squares exist (multiples of $24$)
QUESTION [0 upvotes]: How many positive perfect squares less than $10^6$ are multiples of 24?

I quickly realized:
$$24 = 2^{3}*3*5^0$$
$$10^6 = 2^6 * 5^6*3^0$$
We are finding numbers in the form $24(k^2)$.
But I don't see a way to do this?

REPLY [5 votes]: If a perfect square is a multiple of $24=3\cdot 2^3$, then is a multiple of $144=3^2\cdot 2^4$, too.
Now we have to solve
$$144\le144k^2\le 10^6$$
or
$$12\le 12k\le 10^3$$
which yields
$$1\le k \le 83$$