Relating Translation functor and Jacquet functor via Chan-Wong’s comparison functor

The author would like to thank Professor Bin Xu, Taiwang Deng and Qixian Zhao for patient guidance.

For a real Lie group $G$, take a multiplicative functional $\chi$ over $Z({\mathfrak{g}})$, and then there is a category ${\mathcal{HC}}_{\chi}(G)$ consisting of Harish-Chandra modules over $G$ with generalized infinitesimal character $\chi$. Fix a Cartan subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}$, the complex Lie algebra of $G$, then $\chi$ determines uniquely a Weyl group orbit in ${\mathfrak{h}}^{*}$ according to Harish-Chandra isomorphism. Conversely, a linear functional $\lambda\in{\mathfrak{h}}^{*}$ determines a multiplicative functional $\chi_{\lambda}$ over $Z({\mathfrak{g}})$, and we abbretive ${\mathcal{HC}}_{\chi_{\lambda}}(G)$ by ${\mathcal{HC}}_{\lambda}(G)$.

Chapter VII of [5] defines the translation functor $T_{\lambda}^{\lambda+\mu}:{\mathcal{HC}}_{\lambda}(G)\to{\mathcal{HC}}_{\lambda+\mu}(G)$ for an integral weight $\mu\in{\mathfrak{h}}^{*}$. It is given by $T_{\lambda}^{\lambda+\mu}(X)=P_{\lambda+\mu}(X\otimes F^{\mu})$, where $F^{\mu}$ is a finite dimensional representation of $G$ with extreme weight $\mu$, and $P_{\lambda+\mu}$ is the projection to $(\lambda+\mu)$-primary subspace with respect to $Z({\mathfrak{g}})$.

The graded Hecke algebra ${\mathbb{H}}_{m}$ is generated by the polynomial ring ${\mathbb{C}}[y_{1},\cdots,y_{m}]$ and group algebra ${\mathbb{C}}[S_{n}]$ subject to the relations $s_{i}y_{i}-y_{i+1}s_{i}=1$, and $s_{i}y_{j}-y_{j}s_{i}=0,j\not=i,i+1$, where $s_{i}=(i,i+1)$ are adjacent swaps that form the Coxeter generators of $S_{n}$. Denote the category of finite dimensional modules over ${\mathbb{H}}_{m}$ by ${\mathbb{H}}_{m}{\textrm{-}\mathrm{Mod}}$.

There is an embedding of algebra ${\mathbb{H}}_{m-1}\to{\mathbb{H}}_{m}$ given by $y_{i}\mapsto y_{i+1}$, $s_{i}\mapsto s_{i+1}$. It image is commutative with $y_{1}\in{\mathbb{H}}_{m}$. Then we may define the functor ${\mathrm{Jac}}_{x}:{\mathbb{H}}_{m}{\textrm{-}\mathrm{Mod}}\to{\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$ by taking the generalized $a$-eigenspace with respect to $y_{1}$, where $a$ is any complex number.

Although $G={\mathrm{GL}}(n,{\mathbb{C}})$ has complex structure, we understand it as a real Lie group, and study its representations through $({\mathfrak{g}},K)$-modules.

Denote the Lie algebra of $G$ by ${\mathfrak{g}}_{0}={\mathfrak{gl}}(n,{\mathbb{C}})$, and its complexfication by ${\mathfrak{g}}={\mathfrak{g}}_{0}\otimes_{\mathbb{R}}{\mathbb{C}}$. Then we have the decomposition of real vector space ${\mathfrak{g}}={\mathfrak{g}}_{0}\oplus j{\mathfrak{g}}_{0}$, where $j$ denotes the action of $\sqrt{-1}\in{\mathbb{C}}$ on ${\mathfrak{g}}$. However, it is not a decomposition as complex Lie algebra.

Take the embeddings of complex Lie algebra $\phi^{L},\phi^{R}:{\mathfrak{g}}_{0}\to{\mathfrak{g}}$ defined by

$$\phi^{L}(E)=\frac{1}{2}(E-jiE),\quad\phi^{R}(E)=\frac{1}{2}(\bar{E}+ji\bar{E}).$$

One may check $(\phi^{L},\phi^{R}):{\mathfrak{g}}_{0}\times{\mathfrak{g}}_{0}\to{\mathfrak{g}}$ gives an isomorphism of complex Lie algebra. We may denote the element $\phi^{L}(E)+\phi^{R}(E^{\prime})\in{\mathfrak{g}}$ by $(E,E^{\prime})\in{\mathfrak{g}}_{0}\times{\mathfrak{g}}_{0}$.

Under this coordinate, the image of natural embedding ${\mathfrak{g}}_{0}\to{\mathfrak{g}}_{0}\otimes_{\mathbb{R}}{\mathbb{C}}={\mathfrak{g}}\cong{\mathfrak{g}}_{0}\times{\mathfrak{g}}_{0}$ is given by $\{(E,\bar{E})\mid E\in{\mathfrak{g}}_{0}\}$, and the Cartan involution $\theta$ is given by $(E,E^{\prime})\mapsto(-E^{\prime t},-E^{t})$. Take ${\mathfrak{h}}_{0}\subseteq{\mathfrak{g}}_{0}$ consisting of diagonal matrices, and then ${\mathfrak{h}}={\mathfrak{h}}_{0}\otimes_{\mathbb{R}}{\mathbb{C}}\subseteq{\mathfrak{g}}$ is a Cartan subalgebra, with compact part ${\mathfrak{t}}:={\mathfrak{h}}^{\theta,1}=\{(H,-H)\mid H\in{\mathfrak{h}}_{0}\}$, split part ${\mathfrak{a}}:={\mathfrak{h}}^{\theta,-1}=\{(H,H)\mid H\in{\mathfrak{h}}_{0}\}$.

The maximal compact subgroup of $G$ is $K={\mathrm{U}}(n)$. The diagonal subgroup $H$ decomposes into direct product of torus part $T$ and split part $A$; of course $T=H\cap K$. The Lie algebra of $K$ coincides with $\theta$ fixed points of ${\mathfrak{g}}_{0}$. The complexfications of Lie algebra of $T$, $A$ are exactly ${\mathfrak{t}}$ and ${\mathfrak{a}}$ respectively.

A weight $\lambda\in{\mathfrak{h}}^{*}$ can be denoted by $(\lambda_{L},\lambda_{R})$ under coordinate ${\mathfrak{h}}\cong{\mathfrak{h}}_{0}\oplus{\mathfrak{h}}_{0}$, and $(\mu,\nu)$ under ${\mathfrak{h}}\cong{\mathfrak{t}}\oplus{\mathfrak{a}}$. These two coordinates are related by $\mu=\lambda_{L}-\lambda_{R}$, $\nu=\lambda_{L}+\lambda_{R}$. If $\mu\in{\mathfrak{t}}^{*}$ is integral, $\lambda$ can be lifted to a character of $H=TA$, and to a one dimensional representation ${\mathbb{C}}_{\lambda}$ of the Borel subgroup $B$. Then take the function space ${\operatorname{Ind}}_{B}^{G}{\mathbb{C}}_{\lambda}=\{f\in C^{\infty}(G)\mid f(bg)=\delta(b)^{\frac{1}{2}}\lambda(b)f(g)\},$ where

$$\delta:\begin{pmatrix}b_{1}&&*\\ &\ddots&\\ &&b_{n}\end{pmatrix}\mapsto\prod_{i=1}^{n}|b_{i}|^{2i-1-n}$$

is the modular character of $B$, and $G$ acts by right translation. Denote it by $X(\lambda)$, and call it principle series representation. It also has a $({\mathfrak{g}},K)$-module version by taking $K$-finite subspace. Its infinitesimal character is given by $\lambda$.

Fix a weight $\lambda\in{\mathfrak{h}}^{*}$, and let ${\mathcal{HC}}_{\lambda}$ denote the category of $Z({\mathfrak{g}})$-finite $({\mathfrak{g}},K)$-module of $G$ with generalized infinitesimal character (determined by) $\lambda$. It is an Abelian category, with Grothendieck group denoted by $K{\mathcal{HC}}_{\lambda}$. This group has a basis given by $\{X(w\lambda)\mid w\in W_{\lambda}\}$, where $W_{\lambda}=\{w\in W({\mathfrak{g}},{\mathfrak{h}})\mid w\lambda-\lambda$ is integral $\}$. Under the cordinate $(\phi^{L},\phi^{R})$, $W({\mathfrak{g}},{\mathfrak{h}})=S_{n}\times S_{n}$, and $W_{\lambda}=W_{\lambda_{L}}\times W_{\lambda_{R}}$. Since $\lambda_{L}-\lambda_{R}=\mu$ is integral, $W_{\lambda_{L}}$ coincides with $W_{\lambda_{R}}$ as a subgroup of $S_{n}=W({\mathfrak{g}}_{0},{\mathfrak{h}}_{0})$. Denote it by $W^{\prime}$, and then $W_{\lambda}=W^{\prime}\times W^{\prime}$. According to theorem 2.1 of [4], the basis $\{X(w\lambda)\mid w\in W_{\lambda}\}$ is in bijection to $(\Delta W^{\prime})\backslash(W^{\prime}\times W^{\prime})$, which admits a set of representatives given by $W^{\prime}\times 1\subseteq W^{\prime}\times W^{\prime}$.

There is a natural embedding ${\mathbb{H}}_{m_{1}}\otimes_{\mathbb{C}}{\mathbb{H}}_{m_{2}}\to{\mathbb{H}}_{m_{1}+m_{2}}$. Hence we have the parabolic induction ${\mathbb{H}}_{m_{1}}{\textrm{-}\mathrm{Mod}}\times{\mathbb{H}}_{m_{2}}{\textrm{-}\mathrm{Mod}}\to{\mathbb{H}}_{m_{1}+m_{2}}{\textrm{-}\mathrm{Mod}}$ given by

$$M_{1}\times M_{2}:={\mathbb{H}}_{m_{1}+m_{2}}\otimes_{{\mathbb{H}}_{m_{1}}\otimes_{\mathbb{C}}{\mathbb{H}}_{m_{2}}}(M_{1}\boxtimes M_{2}).$$

Irreducible representations of ${\mathbb{H}}_{m}$ are classified by multi-segements in ${\mathbb{C}}$. By a segement $[a,b]$ with $a,b\in{\mathbb{C}}$, $b-a\in\mathbb{Z}$ we mean the set $\{a,a+1,\cdots b\}$. If $b<a$, then this set is empty.

Irreducible representation of ${\mathbb{H}}_{1}={\mathbb{C}}[y]$ is given by the evaluation $\psi_{c}:y\mapsto c$ at $c\in{\mathbb{C}}$. Then for the segement $\Delta=[a,b]$ with length $m=b-a+1$, there is a representation of ${\mathbb{H}}_{m}$ given by $\psi_{a}\times\cdots\times\psi_{b}$. It has a unique irreducible quotient denoted by ${\mathrm{St}}(\Delta)$. This is a one dimensional representation, with $S_{m}$ acting by ${\mathrm{sgn}}$, and $y_{1},\cdots,y_{m}$ acting by $a,\cdots,b$. For $\Delta=\varnothing$, we define ${\mathrm{St}}(\Delta)$ to be the trivial representation ${\mathbb{C}}$ of ${\mathbb{H}}_{0}={\mathbb{C}}$.

For ${\mathfrak{m}}$ a multi-segement in ${\mathbb{C}}$, enumerate its segements by $\Delta_{1},\cdots,\Delta_{n}$ properly; see subsection 2.4 of [4] for detail. Then ${\mathrm{St}}(\Delta_{1})\times\cdots\times{\mathrm{St}}(\Delta_{n})$ would have a unique irreducible qoutient, denoted by $\mathrm{Speh}({\mathfrak{m}})$. Such representations exhaust all irreducible representations of ${\mathbb{H}}_{m}$. In other words, let $K{\mathbb{H}}_{m}$ denote the Grothendieck group of ${\mathbb{H}}_{m}{\textrm{-}\mathrm{Mod}}$, and then it has a basis given by

$$\{\mathrm{Speh}({\mathfrak{m}})\mid\textrm{length of segements in }{\mathfrak{m}}\textrm{ sum up to }m\}.$$

Moreover, take the graded ring

$$K{\mathbb{H}}=\bigoplus_{m\geqslant 0}K{\mathbb{H}}_{m},$$

with multiplication given by parabolic induction. It is a polynomial ring (over $\mathbb{Z}$), with indeterminates given by $\{{\mathrm{St}}(\Delta)\mid\Delta\subset{\mathbb{C}}$ segements $\}$. We also denote ${\mathrm{St}}(\Delta)$ by $[\Delta]$ when regarding it as an element in the $K$-group.

Let $V$ denote the conjugate standard representation of $G={\mathrm{GL}}(n,{\mathbb{C}})$. This means $V={\mathbb{C}}^{n}$, and $g\in G$ acts on it by left multiplication of the matrix $\bar{g}$.

For $X\in{\mathcal{HC}}_{\lambda}$, $\Gamma_{n,m}(X)=(X\otimes V^{\otimes m})^{K}$ as a vector space. To make it into an ${\mathbb{H}}_{m}$-module, [4] defines an action of ${\mathbb{H}}_{m}$ on $X\otimes V^{\otimes m}$, which is commutative with the action of $G$, and then would preserve $K$-fixed point of $X\otimes V$. Here we present the conceret actions of $s_{k},y_{l}\in{\mathbb{H}}_{m}$ on $v_{0}\otimes v_{1}\otimes\cdots\otimes v_{m}\in X\otimes V^{\otimes m}$ as follows: let $E_{i,j}\in{\mathfrak{g}}_{0}$ denote the $n\times n$ matrix with $1$ on the $(i,j)$-entry, and $0$ on the other entries, and then

(1)		$\displaystyle s_{k}\cdot v_{0}\otimes\cdots\otimes v_{m}=$	$\displaystyle-\sum_{1\leqslant i,j\leqslant n}v_{0}\otimes\cdots\otimes(0,E_{i,j})v_{k}\otimes(0,E_{j,i})v_{k+1}\otimes\cdots\otimes v_{m},$
		$\displaystyle y_{l}\cdot v_{0}\otimes\cdots\otimes v_{m}=$	$\displaystyle\sum_{0\leqslant x<l}\sum_{1\leqslant i,j\leqslant n}v_{0}\otimes\cdots\otimes(0,E_{i,j})v_{x}\otimes\cdots\otimes(0,E_{j,i})v_{l}\otimes\cdots\otimes v_{m}$
			$\displaystyle+\frac{n}{2}v_{0}\otimes\cdots\otimes v_{m}.$

According to [2, equation (2.3)], actions of $s_{k},y_{l}\in{\mathbb{H}}_{m}$ are commutative with $\phi^{R}({\mathfrak{g}}_{0})$; they are commutative with $\phi^{L}({\mathfrak{g}}_{0})$ due to the commutativtity of $\phi^{R}({\mathfrak{g}}_{0})$ and $\phi^{L}({\mathfrak{g}}_{0})$. According to [2, lemma 2.3.3], $s_{k}\in{\mathbb{H}}_{m}$ acts on $X\otimes V^{\otimes m}$ by

$$v_{0}\otimes\cdots\otimes v_{k}\otimes v_{k+1}\otimes\cdots v_{m}\mapsto-v_{0}\otimes\cdots\otimes v_{k+1}\otimes v_{k}\otimes\cdots v_{m}.$$

This follows from [4, Lemma 6.1 and Theorem 6.4].

One of main theorems in [4] is relating (generalized) Bernstein-Zelevinsky functors on the $p$-adic side with tensoring functors on the real side. Our theorem 1.3 could be regarded as a refinement of theirs. We will explain furthur in the remark after lemma 3.1.

There is also a dual statement asserting a natural isomorphism between the following compositions:

$$\Gamma_{n,m-1}\circ T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L}-e_{i},\lambda_{R})}\simeq{\mathrm{Jac}}^{\lambda_{L,i}-\frac{1}{2}}\circ\Gamma_{n,m}.$$

Here, ${\mathrm{Jac}}^{a}:{\mathbb{H}}_{m}{\textrm{-}\mathrm{Mod}}\to{\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$ denotes the functor that takes the generalized $a$-eigenspace with respect to $y_{m}\in{\mathbb{H}}_{m}$, and regards it as an ${\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$ via the natural embedding ${\mathbb{H}}_{m-1}\to{\mathbb{H}}_{m}$, $y_{i}\mapsto y_{i}$, $s_{i}\mapsto s_{i}$. We will discuss it in section 5.

Proof of Theorem 1.3 will be based on the results in [4], and the following lemma.

Here we prove the $K$-group version of theorem 1.3, which asserts commutativity of the following diagram of Abelian groups:

$$\leavevmode\hbox to140.59pt{\vbox to71.41pt{\pgfpicture\makeatletter\hbox{\hskip 70.29338pt\lower-35.75262pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-70.29338pt}{-35.65279pt}\pgfsys@invoke{ }\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip 24.80917pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-20.50363pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\mathcal{HC}_{(\lambda_{L},\lambda_{R})}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 24.80917pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip 0.0pt\hfil&\hfil\hskip 39.11812pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.8126pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\mathbb{H}_{m}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 15.11815pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip 0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 0.0pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip 28.84193pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.53639pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\mathcal{HC}_{(\lambda_{L},\lambda_{R}+e_{i})}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 28.84193pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 0.0pt\hfil&\hfil\hskip 41.45145pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.14594pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{K\mathbb{H}_{m-1}}}$} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip 17.45148pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope \pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{} \pgfsys@moveto{-2.07988pt}{2.39986pt}\pgfsys@curveto{-1.69989pt}{0.95992pt}{-0.85313pt}{0.27998pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85313pt}{-0.27998pt}{-1.69989pt}{-0.95992pt}{-2.07988pt}{-2.39986pt}\pgfsys@stroke\pgfsys@endscope}} }{}{}{{}}{}{}{{}}\pgfsys@moveto{-16.44228pt}{20.50005pt}\pgfsys@lineto{37.1238pt}{20.50005pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.32378pt}{20.50005pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-41.45145pt}{10.64034pt}\pgfsys@lineto{-41.45145pt}{-24.5598pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-41.45145pt}{-24.75978pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{52.8419pt}{12.9348pt}\pgfsys@lineto{52.8419pt}{-24.5598pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{52.8419pt}{-24.75978pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{ }{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.40952pt}{-33.15279pt}\pgfsys@lineto{34.79047pt}{-33.15279pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{34.99045pt}{-33.15279pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$$

This simpler version will be useful for Corollary 3.4, and subsequently Theorem 1.3. It sufficies to prove commutativity for the basis of $K{\mathcal{HC}}_{\lambda}$ given by principle series $\{X(w\lambda_{L},\lambda_{R})\mid w\in W^{\prime}\}$. We may denote $(w\lambda_{L},\lambda_{R})$ by $\lambda^{\prime}=(\lambda_{L}^{\prime},\lambda_{R})$. Take the index set $I=\{k\mid\lambda_{L,k}^{\prime}>\lambda_{R,k}=\lambda_{R,i}\}$. According to lemma 1.1 and lemma 1.2, we have the following in $K{\mathbb{H}}_{m-1}$ if $\Gamma_{n,m}(X(\lambda^{\prime}))\not=0$:

	$\displaystyle{\mathrm{Jac}}_{\lambda_{R,i}+\frac{1}{2}}\circ\Gamma_{n,m}[X(\lambda_{L}^{\prime},\lambda_{R})]=$	$\displaystyle{\mathrm{Jac}}_{\lambda_{R,i}+\frac{1}{2}}([\Delta_{1}]\times\cdots\times[\Delta_{n}])$
	$\displaystyle=$	$\displaystyle\sum_{k\in I}[\Delta_{1}]\times\cdots\times[{}^{-}\Delta_{k}]\times\cdots\times[\Delta_{n}],$

where $\Delta_{k}=[\lambda_{R,k}+\frac{1}{2},\lambda_{L,k}^{\prime}-\frac{1}{2}]$, and ${}^{-}\Delta_{k}=[\lambda_{R,k}+\frac{3}{2},\lambda_{L,k}^{\prime}-\frac{1}{2}]$ is obtained from $\Delta_{k}$ by deleting its beginning. On the other hand, in the Grothendieck group of ${\mathcal{HC}}$ we have

$$T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L},\lambda_{R}+e_{i})}[X(\lambda_{L}^{\prime},\lambda_{R})]=P_{(\lambda_{L},\lambda_{R}+e_{i})}\left(\sum_{k=1}^{n}[X(\lambda_{L}^{\prime},\lambda_{R}+e_{k})]\right)=\sum_{\lambda_{R,k}=\lambda_{R,i}}[X(\lambda_{L}^{\prime},\lambda_{R}+e_{k})]$$

according to lemma 1.4. Note that if $\lambda_{L,k}^{\prime}\geqslant\lambda_{R,k}+1$, then

$$\Gamma_{n,m-1}[X(\lambda_{L}^{\prime},\lambda_{R}+e_{k})]=[\Delta_{1}]\times\cdots\times[{}^{-}\Delta_{k}]\times\cdots\times[\Delta_{n}],$$

and otherwise is $0$. Therefore, we have $\Gamma_{n,m-1}\circ T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L},\lambda_{R}+e_{i})}={\mathrm{Jac}}_{\lambda_{R,i}+\frac{1}{2}}\circ\Gamma_{n,m}$ holds for such $X(w\lambda_{L},\lambda_{R})$. The case when $\Gamma_{n,m}(X(\lambda^{\prime}))=0$ is trivial. We thus conclude the commutativity for $K$-groups.

For any Harish-Chandra module $Z$ of $G$, there is a standard Mackey isomorphism $({\operatorname{Ind}}_{P}^{G}\tau)\otimes Z\cong{\operatorname{Ind}}_{P}^{G}(\tau\otimes Z)$ sending $f\otimes z$ to the function $[g\mapsto f(g)\otimes z]$; see for example [5, Theorem 2.103].

Then we may give

There is a Frobenius reciporcity $({\operatorname{Ind}}_{P}^{G}\tau)^{K}\cong\tau^{K\cap P}$ given by evaluatinng $f\mapsto f(1)$.

As a corollary, there is an isomorphism

(2)

$$\Phi:{\operatorname{Ind}}_{B}^{G}({\mathbb{C}}_{\lambda}\otimes V)^{K}\cong({\mathbb{C}}_{\mu}\otimes V^{\otimes m})^{T}.$$

Note that as $T$-representation,

$$V=\bigoplus_{k=1}^{n}{\mathbb{C}}_{-e_{k}},$$

where the $-e_{k}$-weight space is spanned by $e_{k}\in{\mathbb{C}}^{n}=V$. For $\kappa:\{1,\cdots,m\}\to\{1,\cdots,n\}$, define $e_{\kappa}:=e_{\kappa(1)}\otimes\cdots\otimes e_{\kappa(m)}\in V^{\otimes m}$. Then we know $({\mathbb{C}}_{\mu}\otimes V^{\otimes m})^{T}$ has a basis given by

$$\{1\otimes e_{\kappa}\mid\#\kappa^{-1}(i)=\mu_{i}\}.$$

In particular, this space would be non-zero only when $\mu_{i}\geqslant 0$ and $\mu_{1}+\cdots+\mu_{n}=m$, and in this case it has dimension $\frac{m!}{\mu_{1}!\cdots\mu_{n}!}$. This is the proof for lemma 1.2 given in [4, subsection 5.2].

[4, Section 5] proved $\Gamma_{n,m}$ is compatible with parabolic induction. More explicitly, for $Y_{1}\in{\mathcal{HC}}_{(\mu_{1},\nu_{1})}$, $Y_{2}\in{\mathcal{HC}}_{(\mu_{2},\nu_{2})}$, there is natural isomorphism $\Gamma_{n,m}(Y_{1}\times Y_{2})\cong\Gamma_{n_{1},m_{1}}(Y_{1})\times\Gamma_{n_{2},m_{2}}(Y_{2})$, where $n=n_{1}+n_{2}$, $m=m_{1}+m_{2}$, and $m_{i}$ is the sum of components of $\mu_{i}$. Here we review some details in its proof.

There is a natural embedding $S_{m_{1}}\times S_{m_{2}}\hookrightarrow S_{m}$. Take $S^{m_{1},m_{2}}\subseteq S_{m}$ to be the set of minimal representatives of $(S_{m_{1}}\times S_{m_{2}})\backslash S_{m}$.

Let $\{e_{1},\cdots,e_{n}\}$ be the standard basis of $V={\mathbb{C}}^{n}$, $V_{1}=\operatorname{span}_{\mathbb{C}}\{e_{1},\cdots,e_{n_{1}}\}$ and $V_{2}=\operatorname{span}_{\mathbb{C}}\{e_{n_{1}+1},\cdots,e_{n}\}$. Then for $w\in S^{m_{1},m_{2}}$, there is $\mathcal{X}_{w}\subseteq V^{\otimes m}$ spanned by vectors $v_{1}\otimes\cdots\otimes v_{m}$ where

$$v_{i}\in\left\{\begin{aligned} &V_{1},&\textrm{if }w(i)\in\{1,\cdots,m_{1}\},\\ &V_{2},&\textrm{if }w(i)\in\{m_{1}+1,\cdots,m\}.\end{aligned}\right.$$

Take the parabolic subgroup $P\subseteq G={\mathrm{GL}}(n,{\mathbb{C}})$ consisting of matrices

$$\begin{pmatrix}g_{1}&*\\ &g_{2}\end{pmatrix},\quad g_{i}\in{\mathrm{GL}}(n_{i},{\mathbb{C}}).$$

Then $P\cap K=K_{1}\times K_{2}$, with $K_{i}={\mathrm{U}}(n_{i})\subseteq G_{i}:={\mathrm{GL}}(n_{i},{\mathbb{C}})$. Denote $Y=Y_{1}\boxtimes Y_{2}$, and $P$ acts on it via $P\twoheadrightarrow G_{1}\times G_{2}$. Since $\Gamma_{n,m}(Y_{1}\times Y_{2})=\left({\operatorname{Ind}}_{P}^{G}(Y)\otimes V^{\otimes m}\right)^{K}$ is isomorphic to ${\operatorname{Ind}}_{P}^{G}(Y\otimes V^{\otimes m})^{K}\cong(Y\otimes V^{\otimes m})^{K_{1}\times K_{2}}$, we may denote by $\mathcal{W}_{w}$ the subspace of $\Gamma_{n,m}(Y_{1}\times Y_{2})$ sent to $(Y\otimes\mathcal{X}_{w})^{K_{1}\times K_{2}}$ via this isomorphism .

The first and second statement is Claim 1 and 2 in proof of [4, theorem 5.7]. The third statement is implied by [4, lemma 5.5]; we have also pointed in subsection 1.5 that $w\in S_{m}\subseteq{\mathbb{H}}_{m}$ acts on $x\otimes v_{1}\otimes\cdots\otimes v_{m}\in(X\otimes V^{\otimes m})^{K}=\Gamma_{n,m}(X)$ by

$$x\otimes v_{1}\otimes\cdots\otimes v_{m}\mapsto(-1)^{l(w)}x\otimes v_{w^{-1}(1)}\otimes\cdots\otimes v_{w^{-1}(m)}.$$

Denote the forgetful functor by ${\mathrm{For}}:{\mathbb{H}}_{m}{\textrm{-}\mathrm{Mod}}\to{\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$. There is a natural isomorphism

$${\mathrm{For}}\simeq\bigoplus_{a\in{\mathbb{C}}}{\mathrm{Jac}}_{a}.$$

On the level of objects, it is merely the decomposition into generalized eigenspaces of $y_{1}$.

Consequently, there is a natural isomorphism of functors ${\mathcal{HC}}_{\lambda}\to{\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$:

$${\mathrm{For}}\circ\Gamma_{n,m}\simeq\bigoplus_{a\in{\mathbb{C}}}{\mathrm{Jac}}_{a}\circ\Gamma_{n,m}.$$

We will see later that if $a\not=\lambda_{R,k}+\frac{1}{2},\forall k$, then ${\mathrm{Jac}}_{a}\circ\Gamma_{n,m}$ would be zero. Hence the right hand side of above isomorphism would be a finite sum.

Denote the category of $Z({\mathfrak{g}})$-finite Harish-Chandra modules over $G={\mathrm{GL}}(n,{\mathbb{C}})$ by ${\mathcal{HC}}_{\mathrm{fin}}$. There is a natural transformation of functors ${\mathcal{HC}}_{\lambda}\to{\mathcal{HC}}_{\mathrm{fin}}$ given by

$$T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L},\lambda_{R}+e_{i})}(X)\hookrightarrow X\otimes V.$$

Consequently, we have $\Gamma_{n,m-1}\circ T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L},\lambda_{R}+e_{i})}\Rightarrow\Gamma_{n,m-1}\circ(-\otimes V)$.

The consistency of actions of $s_{k}\in{\mathbb{H}}_{m-1}$ follows either by direct computation like above, or the observation of [2, lemma 2.3.3] as pointed out in subsection 1.5. ∎

Now we obtain the natural transformations of functors ${\mathcal{HC}}_{\lambda}\to{\mathbb{H}}_{m-1}{\textrm{-}\mathrm{Mod}}$

$$\theta^{i}_{a}:\Gamma_{n,m-1}\circ T_{(\lambda_{L},\lambda_{R})}^{(\lambda_{L},\lambda_{R}+e_{i})}\Rightarrow{\mathrm{For}}\circ\Gamma_{n,m}\Rightarrow{\mathrm{Jac}}_{a}\circ\Gamma_{n,m}.$$

We have to prove

We claim that proposition 3.2 holds for each principle series $X(w\lambda),w\in W_{\lambda}$. It sufficies to prove for $X(\lambda)$, as the other $X(w\lambda)$ can be easily incorporated into subsequent argument well.

We have given a filtration $0=X_{0}\subset X_{1}\subset\cdots\subset X_{n}$ of $X(\lambda)\otimes V$ in subsection 2.1, where $X_{k}={\operatorname{Ind}}_{B}^{G}({\mathbb{C}}_{\lambda}\otimes V_{k})$, and $X_{k}/X_{k-1}\cong X(\lambda_{L},\lambda_{R}+e_{k})$.