Therefore the functor $\Hom_{\cH(\wG)}(\cH(\wG),-)\colon \cM(\wG)\to \cH(\wP)\lmod$ is identified with $\Res_\wP^\wG \circ \overline{(-)}$.

Now we conclude using the fact that the image of a non-degenerate module is a non-degenerate module.

We immediately get some expression of the right adjoint of parabolic induction:

The functor $\bfi_{\wL,\wP}^\wG\colon \cM(\wL)\to \cM(\wG)$ is left adjoint to

\[ (-)^N\circ (-)_{nd-\wP}\circ \Res_\wP^\wG\circ \overline{(-)}.\]

Use <ref> and that the right adjoint of the inflation functor $\Res_\wP^\wL\colon \cM(\wL)\to \cM(\wP)$ is $(-)^N$.

The key in the proof of Bernstein's second adjointness is the generalized Jacquet Lemma which was proved first by Casselman for admissible representations and later generalized by Bernstein to all smooth representations.
See <cit.>, [10], and <cit.>.
The same proof works for finite central extensions:

Let $\wK\le \wG$ be an open compact admitting an Iwahori decomposition with respect to $\wP=\wL N$.
Then for any representation $V\in\cM(\wG)$,  the projection

\[V^\wK\to (V_N)^{\wK\cap \wL} \]

is surjective and has a natural (functorial in $V$) section, call it
$s_{\wK}:  (V_N)^{\wK\cap \wL} \to V^\wK$.
Further, for $\wK' \subset \wK$ open compact, both admitting
Iwahori decompositions with respect to  $\wP=\wL N$,
we have the commutative diagram (see <cit.>) :

\[ 	\begin{tikzcd}
(V_N)^{\wK' \cap \wL}	\ar[r,"s_{\wK'}"] \ar[d," e_{\wK \cap \wL}  "'] & 	V^{\wK'}  \arrow[d,"e_{\wK}"]\\
(V_N)^{\wK \cap \wL}	 \ar[r,"s_{\wK}"] &  	V^\wK.
\end{tikzcd}\]

Recall that for a $\wG$-module $V$,  we defined its completion to be

\[ \overline{V}:=\Hom_\wG(\cH(\wG),V) = \lim_\wK V^\wK \]

where the limit is taken over all the open compact subgroups $\wK$ of $\wG$.
The previous theorem is actually equivalent to the following very nice looking statement that is proved for linear groups in <cit.>.
The same proof works for finite central extensions. (The commutativity of the diagram in Theorem
<ref> allows one to construct a map  $\overline{V_{N}} \to \overline{V}$ which is then proved to land inside
the space of $N^-$-invariants.)

If $\wP=\wL N$ is a parabolic with opposite $\wP^- = \wL N^-$ and $V$ is a smooth  representation of $\wG$ then the natural map

\[ \overline{V}^{N^-}\to \overline{V_{N}} \]

is an isomorphism of $\wL$-representations.

In representation theory of $p$-adic groups taking invariants under unipotent groups
is not a well behaved functor but co-invariants are.
The following corollary recovers the lost properties of invariants provided we are willing to work with completed representations.

Using <ref> one can now prove Bernstein's second adjointness theorem easily (we follow the exposition in <cit.> for linear groups).
Other references for the linear case are <cit.>, [10], and [16].

The functor $\bfi_{\wL,\wP}^\wG$ is left adjoint to $\bfr_{\wL,\wP^-}^\wG$.

We have the following natural isomorphisms:

\begin{align*}
\Hom_\wG(\bfi_{\wL,\wP}^\wG(V),W) &
= \Hom_\wG(\ind_\wP^\wG \Res_\wP^\wL (\delta_P^{1/2}\otimes V),W)  & \text{ by definition}\\
&=\Hom_\wG(\cH(\wG)\otimes_{\cH(\wP)}\Res_\wP^\wL(\delta_P^{-1/2}\otimes V),W) & \text{by \cref{P:ind_P^G as tensor product}}\\
& = \Hom_\wP(\Res_\wP^\wL(\delta_P^{-1/2}\otimes V),\overline{W}) & \text{by \cref{P:right adjoint to ind_P^G}}\\
& = \Hom_\wL(\delta_P^{-1/2}\otimes V,\overline{W}^N) & \Res_\wP^\wL \dashv (-)^N\\
& = \Hom_\wL(V,\delta_P^{1/2}\otimes \overline{W_{N^-}})&\text{by \cref{C:invar and coinvar are isomorphic}}\\
& = \Hom_\wL(V,\delta_P^{1/2} \otimes W_{N^-})& \text{ by \cref{R:non-deg is right adjoint}}\\
& = \Hom_\wL(V,\bfr_{\wL,\wP^-}^\wG(W)) & \text{ by definition.}
\end{align*}

Since every functor that admits an exact right adjoint preserves projective objects, we deduce the rather non-trivial fact:

The functor $\bfi_{\wL,\wP}^\wG\colon \cM(\wL)\to \cM(\wG)$ sends projective objects to projective objects.

§ BLOCKS AS MODULE CATEGORIES

In this section we describe each block $\cM(\wG)_\fs$, $\fs\in\cB(\wG)$, as the category of modules over some algebra $\cR_\fs$ and prove some basic homological properties of this algebra.

The key results of this section are <ref> and <ref>.

§.§ Cuspidals

Fix $(\wL,\rho)$  a cuspidal data and denote by $\fs\in\cB(\wG)$ its equivalence class (conjugation and inertia).

Define the following representation of $\wL$:

\[ \Pi_{[\rho]}:=\ind_{\wLo}^\wL (\rho\mid_{\wLo}) =\rho \otimes  \ind_{\wLo}^\wL (\bC),
\]

where we recall that $\wLo$ is the subgroup of $\wL$ generated by all compact subgroups.

The representation $\Pi_{[\rho]}$ is a finite type projective generator in $\cM(\wL)_{[\rho]}$.

First we show the projectivity.
Since $\wLo\le\wL$ is an open subgroup, the functor $\ind_{\wLo}^\wL$ is also left adjoint to $\Res_{\wLo}^\wL$ (see <ref>) which is exact, hence $\ind_{\wLo}^\wL$ preserves projective objects.
It is therefore enough to show that $\rho|_\wLo$ is projective which follows immediately from  Harish-Chandra's <ref> and the fact that compact representations are projective by <ref>.

Let us now show that $\Pi_{[\rho]}$ is a generator of $\cM(\wL)_{[\rho]}$.
Let $\pi$ be an arbitrary representation in $\cM(\wL)_{[\rho]}$.
We must show that there's a non-zero morphism $\Pi_{[\rho]}\to \pi$.
Recall (paragraph before <ref>) that the category $\cM(\wL)_{[\rho]}$ is defined as consisting of those smooth representations of $\wL$ all  whose irreducible  subquotients belong to $[\rho]$.
Since  $\Pi_{[\rho]}$ is a projective representation of $\wL$, it is  enough to prove that
an irreducible representation belonging to $\cM(\wL)_{[\rho]}$, thus  isomorphic to $\chi\rho$ for some unramified character $\chi\in\cX(\wL)$, appears as a quotient of $\Pi_{[\rho]}$.
By the adjunction <ref>, we have

\[ \Hom_{\wL}(\Pi_{[\rho]},\chi\rho) = \Hom_\wLo(\rho\mid_{\wLo},\chi\rho\mid_\wLo), \]

and the latter is non-zero as it contains the identity.

In order to show that
is of finite type, one notices that $\wLo\backslash \wL$ is a discrete group and leads to a basis of $\ind_\wLo^\wL(\rho\mid_\wLo)$ indexed by these cosets and a basis of $\rho$.
The action of $\wL$ permutes this basis and hence, by choosing a finite generating set of $\rho\mid_\wLo$, we obtain a finite generating set of $\Pi_{[\rho]}$ as an $\wL$-representation, proving that $\Pi_{[\rho]}$ is of finite type.

We put $\cR_{[\rho]}:=\End_\wL(\Pi_{[\rho]})$. Generalities from
category theory (see <ref>), give us the following corollary:

We have an equivalence of categories

\begin{align*}
\cM(\wL)_{[\rho]} &\longrightarrow \rmod\cR_{[\rho]}\\
\pi &\mapsto \Hom_\wL(\Pi_{[\rho]},\pi).\notag
\end{align*}

Below we give a description of the algebra $\cR_{[\rho]}$ by generators and relations.

Recall that $\cG_\rho\subset  \Hom_\gr(\Lambda,\bC^\times) = \cX(\wL)$ denotes the stabilizer of $\rho$, where $\Lambda = \wL/\wLo$, i.e.,  $\cG_\rho = \{\chi \in  \Hom_\gr(\Lambda,\bC^\times) | \rho \otimes \chi \simeq \rho\}$.
The group $\cG_\rho$ naturally acts on the complex torus  $\cX(\wL)$
by translations, and hence also on the
algebra of regular functions on $\cO(\cX(\wL)) = \bC[\Lambda]:= \cA$ which we denote by $\chi(f)= {}^\chi f$ for
$\chi \in \cG_\rho$. By Schur's lemma, the isomorphism $\rho \otimes \chi \simeq \rho$ defines a unique map – up to scalars –
from the space
underlying $\rho$ to itself, hence provides us with a 2-cocycle  $ c \in H^2(\cG_\rho, \bC)$, which in turn defines a
twisted group algebra $\bC[\cG_{\rho},c]$, the algebra generated by the intertwining operators from $\rho$ to itself
through the isomorphisms $\rho \otimes \chi \simeq \rho$.

The algebra $\cR_{[\rho]}$ has the following presentation:

* As a vector space $\cR_{[\rho]} = \cA\otimes \bC[\cG_{\rho},c]$,

* $\cA$ and $\bC[\cG_{\rho},c]$ are subalgebras,

* $f b_\chi = b_\chi {}^\chi f$ for all $f\in\cA$ and $\chi\in \cG_{\rho}$, with
$b_\chi = 1 \otimes \chi \in  \cA\otimes \bC[\cG_{\rho},c] =\cR_{[\rho]}$.

The proof is fairly elementary but we will give a sketch.

First notice that by adjunction, we have a natural isomorphism of vector spaces

\begin{align}\label{Eq:R_rho as v space}
\cR_{[\rho]} = \End_\wL(\ind_\wLo^\wL(\rho|_\wLo)) \simeq \End_\wLo(\rho|_\wLo)\otimes \cA.
\end{align}

Since	$\End_\wL(\ind_\wLo^\wL(\bC)) = \cA$, and since $\ind_\wLo^\wL(\rho|_\wLo)\simeq \rho\otimes \ind_\wLo^\wL(\bC)$, this exhibits $\cA$ as a subalgebra of $\cR_{[\rho]}$ and moreover it shows that (<ref>) is an isomorphism of right $\cA$-modules.

Second, notice that $\Lambda=\wL/\wLo$ acts on $\End_\wLo(\rho|_\wLo)$ and the action factors through the finite quotient $\wL/Z(\wL)\wLo$.
Moreover the action is diagonalizable and the eigenvalues (i.e., characters of $\Lambda$) that appear are, by definition, exactly the elements of $\cG_\rho$.
We deduce a canonical isomorphism of vector spaces

\[ \End_\wLo(\rho|_\wLo) \simeq \bC[\cG_\rho]. \]

If $b_\chi,b_\mu\in \End_\wLo(\rho|_\wLo)$ correspond to $\chi,\mu\in\cG_\rho$ through the previous isomorphism, then their product belongs to the eigenspace corresponding to $\chi\mu$, i.e., we have $b_\chi b_\mu = c(\chi,\mu)b_{\chi\mu}$ for some non-zero $c(\chi,\mu)\in\bC^\times$.
Associativity plus unity implies that $c(-,-)$ is a two cocycle of $\cG_\rho$ with values in $\bC^\times$.
This allows us to construct a twisted group algebra $\bC[\cG_\rho,c]$ and in turn provides us with an isomorphism of algebras

\[ \End_\wLo(\rho|_\wLo) \simeq \bC[\cG_\rho,c].\]

The last step consists in showing how the subalgebra $\cA$ and $\bC[\cG_\rho,c]$ interact inside $\cR_{[\rho]}$.
This is done by writing down explicitly the isomorphism in (<ref>).

Recall (see <cit.>) that an Azumaya algebra $R$ over a commutative ring $Z$ with identity
is a $Z$-algebra such that for some faithfully flat extension of commutative rings $Z\hookrightarrow Z'$ we have $R\otimes_ZZ'\simeq M_n(Z')$.
Equivalently, $R$ is a projective module of constant rank over $Z$ and the multiplication map $R\otimes_Z R^\op \to \End_Z(R)$ is an isomorphism of rings.

In particular, one deduces in a straightforward way the following proposition
from the above presentation of $\cR_{[\rho]}$.

The natural inclusion $\cA^{\cG_\rho}\hookrightarrow \cR_{[\rho]}$ identifies $\cA^{\cG_\rho}$ with the center $Z(\cR_{[\rho]})$.
In particular $Z(\cR_{[\rho]})$ is isomorphic to a Laurent polynomial algebra.
Moreover $\cR_{[\rho]}$ is an Azumaya algebra over its center.

For a different proof, see <cit.>.
Note that our algebra $\cR_{[\rho]}$ is a matrix algebra over the algebra $E_G$ of [11].

The next lemma is easy and is valid for any Azumaya algebra (see <cit.>).

Let $R$ be an Azumaya algebra over its center $Z$ which is a noetherian ring.
Then the trace map $\tr \colon R \to Z$ provides an isomorphism of $R$-bimodules

\[ R\to \Hom_Z(R,Z).\]

First,  this is obvious for $R=M_n(Z)$.
Using faithfully-flat descent, one first defines the trace map unambiguously for any Azumaya algebra
$R$        and then sees that it provides a non-degenerate symmetric pairing $R\otimes_Z R\to Z$ which leads to the isomorphism asserted in  the statement of the lemma.

Applying this to our algebra $\cR_{[\rho]}$ with center $\cZ_{[\rho]}:=Z(\cR_{[\rho]})$ which is a Laurent polynomial ring, we get the following corollary (see (<ref>)):

There is an isomorphism of $\cR_{[\rho]}$-bimodules

\[\cR_{[\rho]}\simeq \Hom_{\cZ_{[\rho]}}(\cR_{[\rho]},\cZ_{[\rho]}).\]

In other words, $\cR_{[\rho]}$ satisfies condition (FsG) in Definition <ref>.

Let us now apply what we have learned about $\cR_{[\rho]}$ to prove the following proposition which plays a crucial role in the study of the homological duality functor (see <ref>).

Recall that for $\rho$ a cuspidal representation of $\wL$, we defined $d(\rho)$ to be the split rank of the center of $L$ (which is a Levi subgroup of $G$).

For $V$, a $\cR_{[\rho]}$-module which is finite dimensional over $\bC$,  we have

\begin{align*}
\Ext^i_{\cR_{[\rho]}}(V,\cR_{[\rho]}) = 0 \text{ for all }i\neq d(\rho).
\end{align*}

For simplicity we will put $\cR = \cR_{[\rho]}$ and $\cZ:=Z(\cR_{[\rho]})$ in this proof.
The forgetful functor

\begin{align*}
F\colon \rmod\cR\to  \rmod\cZ
\end{align*}

can also be written as $F = -\otimes_{\cR}\cR$ where ${}_\cR\cR_\cZ$ is viewed as an $\cR\mhyp\cZ$-bimodule.
As such, $F$ has a right adjoint $F':=\Hom_\cZ(\cR_\cZ,-)$.

Since $\cR$ is an Azumaya algebra (<ref>), it is a projective $\cZ$-module, hence  both $F$ and $F'$ are exact functors.
It follows that the adjunction extends to Ext groups.
Applying <ref> we therefore get

\[ \Ext^i_\cR(V,\cR) = \Ext^i_\cR(V,F'(\cZ)) = \Ext^i_\cZ(V,\cZ),\]

and now we conclude the proof of the proposition by a well-known result in commutative algebra
since $\cZ$ is a Laurent polynomial algebra (by <ref>)
of Krull dimension $d(\rho)$ and $V$ is a
$\cZ$-module which is a
finite dimensional vector space  over $\bC$.

The statement holds with the same proof if we only assume that $\cR_{[\rho]}\simeq\Hom_{\cZ_{[\rho]}}(\cR_{[\rho]},\cZ_{[\rho]})$ as right $\cR_{[\rho]}$-modules.

§.§ Non-cuspidal blocks: a projective generator

Moving on to the block in $\wG$ corresponding to $\fs=[\wL,\rho]\in\cB(\wG)$, let us define the representation

\[ \Pi_\fs:=\oplus_{\wP} \bfi_{\wL,\wP}^\wG(\Pi_{[\rho]}) = \oplus_{\wP}\bfi_{\wL,\wP}^{\wG} (\ind_{\wLo}^{\wL}(\rho|_{\wLo}))\]

where the sum ranges over all parabolics $\wP$ with Levi $\wL$ (it is a finite sum).

Actually it would be enough to consider $\bfi_{\wL,\wP}^\wG(\Pi_{[\rho]})$ for a single parabolic as this module turns out to be (non-canonically) independent of the parabolic containing $\wL$.
See  <cit.>) for a proof as well as
<cit.> for some more details.
However, we will not need it.
One advantage of taking the direct sum instead of a single parabolic induction is that it makes the isomorphism from <ref> canonical.

The module $\Pi_\fs$ is a finite type projective-generator of $\cM(\wG)_\fs$.

The projectivity follows at once as a consequence of the second-adjointness <ref> together with <ref>: the functor $\bfi_{\wL,\wP}^{\wG}$ has an exact right adjoint, hence it preserves projectives and $\Pi_{[\rho]}$ is projective.
Since parabolic induction preserves finite type (see <ref>) we have that $\Pi_\fs$ is also finitely generated.

Let $\pi\in \cM(\wG)_\fs$ be a  non-zero representation of $\wG$.
We need to show that $\Hom_{\wG}(\Pi_\fs,\pi)$ is non zero and for that,
since $\Pi_\fs$ is projective, it is enough to consider the case $\pi$ irreducible.
By definition of the block $\cM(\wG)_\fs$, $\bfr_{\wL,\wP^-}^\wG(\pi)$ is non-zero in the block
The second-adjointness <ref> and the fact that $\Pi_{[\rho]}$ is a generator of $\cM(\wL)_{[\rho]}$ (see <ref>) imply that

\[ \Hom_\wG( \bfi_{\wL,\wP}^\wG(\Pi_{[\rho]}),\pi) = \Hom_\wL(\Pi_{[\rho]}, \bfr_{\wL,\wP^-}^\wG(\pi))\neq 0 \]

and hence $\Hom_\wG(\Pi_\fs,\pi)\neq 0$.

An abelian category admitting all coproducts and having a finitely generated (compact) progenerator is equivalent to the category of right modules over its endomorphism ring (see <ref>).
Putting $\cR_\fs:=\End_{\wG}(\Pi_\fs)$, we thus deduce the following:

The functor

\begin{align*}
\cM(\wG)_\fs &\to \rmod\cR_\fs\\
V&\mapsto \Hom_{\wG}(\Pi_\fs,V)
\end{align*}

is an equivalence of categories sending finite length representations of $\wG$ in
to $\cR_\fs$-modules which are finite dimensional over $\bC$.

§.§ Center of $\cM(\wG)_\fs$

This section plays no role in this work but we record it for completeness.
We give the description of the center of the algebra $\cR_\fs$ for $\fs = [\wL,\rho]\in\cB(\wG)$.
In turn this determines the center of the block $\cM(\wG)_\fs$ (see <ref>, <ref> and <ref>).

Recall that $\cX(\wL)$ acts on the irreducible cuspidals $\Irr(\wL)_\cs$ and we denoted by $\cG_\rho$ the stabilizer of $\rho$.
Denote by $W_{[\rho]}:=\Stab_{W_{\wL,\wG}}([\rho])$,  the stabilizer of the inertia class $[\rho]$ in the relative Weyl group $W_{\wL,\wG}$.

Recall also the $\bZ$-lattice $\Lambda(\wL) = \wL/\wLo = \Lambda(L) = L/\Lo$.
The characters of $\Lambda(\wL)$ form the torus $\cX(\wL)$  whose ring of regular functions is identified with the group algebra of $\Lambda(\wL)$:

\[ \bC[\Lambda(\wL)] = \cO(\cX(\wL)). \]

The linear case of the following theorem is due to <cit.> but the proof works also for finite central extensions.
Another reference is <cit.>.

The algebra $\cR_\fs$ contains the Laurent polynomial algebra $\cO(\cX(\wL)/\cG_\rho)$ as a subalgebra and it is finite as a left (or right)-module over it.
Moreover, the center of $\cR_\fs$ is nothing but

\[ \cZ(\cR_\fs) =\cO(\cX(\wL)/\cG_\rho)^{W_{[\rho]}}. \]

Since $\cZ(\cR_\fs)$ has no non-trivial idempotents, we immediately get the following corollary.

The category $\cM(\wG)_\fs$ is indecomposable.

§.§ Revisiting Howe's theorem

Since Bernstein's center plays a prominent role in the theory, it seems worthwhile mentioning the following applications whose proofs are parallel to the linear case. This section is only for completeness, it is not used in the sequel.

We say that a smooth representation $V$ of $\wG$ is $\cZ(\wG)$-admissible (where $\cZ(\wG)$ is the Bernstein
center of the category of smooth representations of $\wG$)
if for any open compact $\wK\le \wG$,  the $\cZ(\wG)$-module $V^{\wK}$ is finitely generated.
We have

A smooth representation $V$ of $\wG$ is of finite type if and only if it is $\cZ(\wG)$-admissible and it has only finitely many non-zero Bernstein components.

We say that a representation is $\cZ(\wG)$-finite if the action of $\cZ(\wG)$ on it factors through an ideal of finite codimension in  $\cZ(\wG)$.

The following is a generalization of Howe's theorem.
Let $V$ be a smooth representation of $\wG$.
Any two of the following properties implies the third, and then $V$ is of finite length:

* $V$ is of finite type,

* $V$ is admissible,

* $V$ is $\cZ(\wG)$-finite.

Howe's theorem says that (1)+(2) imply finite length and hence by Schur's lemma we have (3).

Assume (1)+(3). <ref> implies that $V$ is $\cZ(\wG)$-admissible. Hence for any open compact $\wK\le \wG$ the invariants $V^\wK$ are finitely generated over $\cZ(\wG)$ and now (3) implies that $V^\wK$ is finite dimensional.

Assume (2)+(3). Clearly $V$ is $\cZ(\wG)$ admissible and since the action of $\cZ(\wG)$, a unitary ring, factorizes through a finite codimension ideal we get that $V$ has only finitely many non-zero Bernstein components. We conclude by <ref>.

§ HOMOLOGICAL DUALITY

§.§ Vanishing

Here we prove the vanishing part of our main result, namely <ref>(<ref>),
following the strategy in [7].

The main ingredients are <ref> on Bernstein's decomposition, Bernstein's second adjoint <ref>,
and the vanishing result from <ref>.

$\cM(\wG)_{fg}$ the full subcategory of smooth representations of $\wG$ which are finitely generated.
It is still an abelian category and the parabolic induction and restriction functors preserve it.
The homological duality is defined as the following functor between derived categories

\begin{align}
D_h\colon &\cD^b_{fg}(\cM(\wG))\to \cD^b_{fg}(\cM(\wG))^\op\\
& \pi \mapsto \RHom_\wG(\pi,\cH(\wG))\notag
\end{align}

where $\cH(\wG)$ denotes the Hecke algebra of $\wG$.
The structure of left $\wG$ representation is through the right action of $\cH(\wG)$ on itself and through the involution $f\mapsto \breve{f}\colon \cH(\wG)\to \cH(\wG)$ defined by $\breve{f}(g):=f(g\inv)$.

* Notice that $D_h$ lands indeed in the bounded derived category because the category of smooth representations $\cM(\wG)$ has finite global dimension (<ref>).

* One could also define $D_h$ without the assumption on finitely generated but then $D_h$ would land outside smooth modules, indeed in the category of all $\cH(\wG)$-modules.
We could come back to smooth modules simply by taking the smooth part (see the paragraph after <ref>).

The following theorem is the most important property of the homological duality functor and is due to Bernstein.
Fix $\fs = [\wL,\rho]\in\cB(\wG)$, a cuspidal datum.

If  $\pi\in\cM(\wG)_\fs$ is of finite length, then $D_h(\pi)$ has cohomology only in degree $d(\fs)$.

(following <cit.>)
We need to show that $\Ext^i_\wG(\pi,\cH) = 0$ for $i\neq d(\fs)$, where we put for short $\cH = \cH(\wG)$.

Decompose the representation $\cH(\wG)$ according to Bernstein's decomposition,
Clearly only the component $\cH(\wG)_\fs$ is important to us.
Moreover, since $\cH(\wG)$ is a projective representation (<ref>),
and $\Pi_\fs$ is a projective generator of $\cM(\wG)_\fs$ (see <ref>), it is enough to show:

\begin{align}\label{Eq:Ext vanishing with Pi_fs}
\Ext^i_\wG(\pi,\Pi_\fs)=0,\text{ for all }i\neq d(\fs).
\end{align}

Recall from <ref> that $\Pi_\fs = \bfi_{\wL,\wP}^{\wG}(\Pi_{[\rho]})$.
Since parabolic induction and restriction are exact functors, the Frobenius adjunction gives

\begin{align}\label{Eq:Ext with Pi_fs equal with Pi_rho}
\Ext^i_\wG(\pi,\Pi_\fs) = \Ext^i_\wL(\bfr_{\wL,\wP}^\wG(\pi),\Pi_{[\rho]}).
\end{align}

The representation $\pi$ is of finite length, hence by <ref> the same is true of $\bfr_{\wL,\wP}^\wG(\pi)$.
Through the categorical equivalence $\cM(\wL)_{[\rho]}\simeq \rmod\cR_{[\rho]}$ (see <ref>), the vanishing of (<ref>) follows from

\[ \Ext^i_{\cR_{[\rho]}}(V,\cR_{[\rho]}) = 0, \text{ for all }i\neq d(\rho)=d(\fs), \]

for all finite dimensional $\cR_{[\rho]}$-modules $V$.
This was proved in <ref>.

Given $\pi\in\cM(\wG)_\fs^\finl$ a finite length representation in a given block, we will denote by $\bD_h(\pi)$ the representation $H^{d(\fs)}(D_h(\pi))$.

§.§ Interaction with induction and restriction

The objective of this section is to investigate how does the (full, derived) homological duality commute with parabolic induction and restriction.
We follow the proof of the linear case from <cit.>.

Let $\wP = \wL N$ be a parabolic with Levi decomposition in $\wG$.
We have the following natural isomorphisms of functors when restricted to the bounded derived category of finitely generated smooth $\wG$-modules $\cD^b_{fg}(\cM(\wG))$:

* $D_h \bfi_{\wL,\wP}^\wG \simeq \bfi_{\wL,\wP^-}^\wG D_h$,

* $D_h \bfr_{\wL,\wP}^\wG \simeq \bfr_{\wL,\wP}^\wG D_h$.

To shorten the notation, we write $\bfi_\wP$ and $\bfr_\wP$ for the parabolic induction and restriction functors.

Since all objects in $\cM(\wG)_{fg}$ admit a finite resolution by finitely generated
projective objects, it is enough to define and prove the required isomorphisms for finitely
generated projective objects.
Therefore, using the second adjointness and the Frobenius reciprocity,
we see that the isomorphisms in part (1) and in part (2) of the proposition are equivalent to proving natural isomorphisms

\begin{align}
\bfi_{\wP^-} \Hom_\wL(V,\cH(\wL)) &\simeq \Hom_\wL(V,\bfr_{\wP^-}(\cH(\wG))),\label{Eq:ind Hom}\\
\bfr_\wP\Hom_\wG(W,\cH(\wG)) & \simeq \Hom_\wG(W,\bfi_\wP(\cH(\wL))),\label{Eq:res Hom}
\end{align}

for $V\in\cM(\wL)$ and $W\in\cM(\wG)$, both projective, and finitely generated representations, and where
we have considered $\cH(\wL)$ as an $\wL\times \wL$ module and $\cH(\wG)$ as a $\wG\times \wG$-module.

For the proof of these isomorphisms,  we begin by noticing
the following identifications of $\wL\times\wG$-modules:

\begin{align}\label{Eq:ind of H(L)}
(\id\times\bfi_\wP)(\cH(\wL)) \simeq \Ind_{ \wL \times \wP}^{\wL \times \wG}(\cH(\wL)\delta^{-1/2})   \simeq \delta^{1/2} \otimes \cC^\infty_c(\wG/N)
\simeq (\id\times\bfr_\wP)(\cH(\wG)),
\end{align}

where in the first appearance, $\delta$ is a character of $\wL \times \wP$ trivial on $\wL$, whereas in the second appearance, $\delta$ is a character of $\wL \times \wG$ trivial on $\wG$; this also explains why
the power of delta changes sign since $\cH(\wL) = \ind^{\wL \times \wL}_{\Delta(\wL)} \bC$.

The following is an easy but crucial observation valid for any finitely generated
smooth representation $V$ of $\wL$ (we will apply this to finitely generated projective modules
as noted earlier):

\begin{align} \label{Eq:easy crucial obs}
\bfi_\wP\Hom_\wL(V, \cH(\wL)) &\simeq \Hom_\wL(V,(\id\times\bfi_\wP) (\cH(\wL)), \end{align}

as representations of $\wG$ where
in $\Hom_\wL(V,\cH(\wL))$,  both $V$ and $\cH(\wL)$ are considered as left $\wL$-modules,
thus $\Hom_\wL(V,\cH(\wL))$ is a right $\wL$-module through the right $\wL$-action on $\cH(\wL)$; the
$\wL \times \wG$-representation $(\id\times\bfi_\wP) (\cH(\wL)) = \bfi_\wP^{\wG} (\cH(\wL))$ has
$\wL$ action through the left action
of $\wL$ on  $\cH(\wL)$, and a right  $\wG$ action.
To see the isomorphism in equation (<ref>), note that
by the definition of an induced representation, both sides of (<ref>)
give rise to  functions
$F\colon \wG \times V \rightarrow \cH(\wL)$ which are linear maps $V \rightarrow \cH(\wL)$ when restricted
to any $g \in \wG$, and satisfy:

* $F(g, \ell v) = \ell F(g,v)$, for all $g\in \wG, \ell \in \wL, v \in V$.

* $F(gp, v) = F(g,v)\cdot \ell$ where $p \in \wP = \wL N$ has the form $p=\ell n$, with $\ell \in \wL$, and $n \in N$.

Further, such an $F$ arises from the left hand side of the isomorphism in (<ref>) if and only if
the corresponding map $\wG \rightarrow \Hom (V, \cH(\wL))$ is locally constant on $\wG$, whereas
such an $F$ arises from the right hand side of the equality in eq. (<ref>) if and only if for each
$v \in V$, $F(-,v)$  is a locally constant map from $\wG$ to $\cH(\wL)$. Thus the functions $F$ which arise from the left hand side of equation (<ref>) are always contained in those which arise from the right hand side, and the converse holds if  $V$ is finitely generated over $\wL$,
say by $v_1,\cdots, v_n$. Then assuming that each of the functions $F(-,v_i)$ are constant in a neighborhood $U(g_0)$
of a fixed $g_0\in \wG$, then by equation 1. above,  $F(g,\ell v_i) = \ell F(g, v_i)=\ell F(g_0, v_i)$
for all $g \in U(g_0)$, $\ell \in \wL$,
hence $F(g,v) = F(g_0,v)$ for all $g \in U(g_0)$, $v \in V$. Now the isomorphism (<ref>) (applied to
$\wP^-$ in place of $\wP$) together with (<ref>) proves the isomorphism (<ref>), and hence the isomorphism
in part 1. of the proposition.

Similarly for any finitely generated smooth, projective, representation $W$ of $\wG$ we have

\[ \bfr_\wP\Hom_\wG(W,\cH(\wG))\simeq \Hom_\wG(W,

as $\wL$-modules and naturally in $W$. This assertion is equivalent to proving that the normalized Jacquet module of
$\Hom_\wG(W,\cH(\wG))$ (considered as a right $G$-module) is the same as  $\delta^{1/2}\Hom_\wG(W,\cC^\infty_c(\wG/N) )$ which is equivalent to proving that:

* the natural map from
$\Hom_\wG(W,\cH(\wG))$ to   $\Hom_\wG(W,\cC^\infty_c(\wG/N) )$ is surjective. This is a consequence of $W$ being projective.

* The kernel of the natural map $\Hom_\wG(W,\cH(\wG)) \to \Hom_\wG(W,\cC^\infty_c(\wG/N) )$ consists of
$\Hom_\wG(W,\cH(\wG)) [N]$
where for any smooth representation ${\mathcal W}$ of $N$,

\[{\mathcal W}[N] = \{n\cdot w -w| w \in{\mathcal W}\}
=  \{v \in {\mathcal W}| \int_{N_i}n\cdot v = 0 \}\]
where $N_i$ is some compact open subgroup of $ N$ depending on
$v \in {\mathcal W}$. It follows that

\[\Hom_\wG(W,\cH(\wG)) [N] = \Hom_\wG(W,\cH(\wG) [N]),\]

using that $ \Hom_\wG(W,\cH(\wG))$ is a smooth representation of $N$ which is the case as $W$ is a finitely generated $\wG$-module.

Applying the functor $\Hom_\wG(W,-)$ to the exact sequence

\[ 0 \to \cH(\wG)[N] \to \cH(\wG) \to \cC^\infty_c(\wG/N) \to 0,\]

and noting that $W$ is projective, we have the exact sequence

\[ 0 \to  \Hom_\wG(W,\cH(\wG) [N]) \to  \Hom_\wG(W,\cH(\wG)) \to \Hom_\wG(W, \cC^\infty_c(\wG/N)) \to 0,\]

proving the assertion that the Jacquet module of
$\Hom_\wG(W,\cH(\wG))$, considered as a right $\wG$-module, is the same as  $\Hom_\wG(W,\cC^\infty_c(\wG/N) )$.
This completes the proof of the isomorphism
in (<ref>), and hence part 2. of the proposition.

§.§ On projective generators

In order to better understand the homological duality functor $D_h$, we will compute its value on the projective generators $\Pi_{[\rho]}\in\cM(\wG)_{[\rho]}$ and $\Pi_\fs\in \cM(\wG)_\fs$ from <ref>.
The answer is as nice as it could possibly be.
Then we will use this computation to describe the functors $(-)^\vee$ and $D_h$ on the $\cR_\fs$-side: we obtain the contragredient, resp. homological, duality for $\cR_\fs$.

Recall that if $\rho\in\cM(\wG)$ is irreducible cuspidal, then $\Pi_\rho$ was defined to be $\ind_{\wGo}^\wG(\rho|_\wGo)$.
Since a cuspidal representation is compact modulo center by Harish-Chandra's theorem, the following is a restatement of <ref>:

If $\rho\in\cM(\wG)$ is irreducible cuspidal, then

\[ D_h(\Pi_{[\rho]})\simeq \Pi_{[\rho^\vee]}.\]

Let $\fs = (\wL,\rho)\in\cB(\wG)$ be a cuspidal datum (up to inertia and conjugation).
The projective generator $\Pi_\fs\in\cM(\wG)_\fs$ was defined in <ref> as

\[ \Pi_\fs = \oplus_{\wP} \bfi_{\wL,\wP}^\wG \Pi_{[\rho]}.\]

We denote by $\fs^\vee = (\wL,\rho^\vee)$ the contragredient cuspidal datum (it could be the same as $\fs$ but in general it is not).

With the above notation we have
$D_h(\Pi_\fs)\simeq \Pi_{\fs^\vee}$.

Use the previous lemma together with $D_h \bfi_{\wL,\wP}^\wG\simeq \bfi_{\wL,\wP^-}^\wG D_h$ (see <ref>) to get the following isomorphisms

\begin{align*}
D_h(\Pi_\fs) & = \oplus_{\wP} D_h(\bfi_{\wL,\wP}^\wG(\Pi_{[\rho]}))\\
&\simeq \oplus_{\wP} \bfi_{\wL,\wP^-} D_h(\Pi_{[\rho]})\\
& \simeq \oplus_{\wP}\bfi_{\wL,\wP^-}\Pi_{[\rho^\vee]}\\
& = \Pi_{\fs^\vee}.\qedhere
\end{align*}

Homological duality restricts to a functor

\[D_h\colon \cD^b(\cM(\wG)_\fs)\to \cD^b(\cM(\wG)_{\fs^\vee})^\op\]

which is moreover involutive when restricted to finitely generated modules.

The functor $D_h$ sends the projective generator $\Pi_\fs$ of $\cM(\wG)_\fs$ to the projective generator $\Pi_{\fs^\vee}$ of $\cM(\wG)_{\fs^\vee}$.
The duality statement for finitely generated follows from <ref>.

Thanks to the vanishing result for finite length modules (<ref>), we can deduce

The functor $D_h$ restricts to an involution

\[ \bD_h\colon \cM(\wG)_\fs^\finl \to (\cM(\wG)_{\fs^\vee}^\finl)^\op.\]

The homological duality induces an anti-isomorphism of algebras

\[ \cR_\fs\simeq \cR_{\fs^\vee},\]

or equivalently, an isomorphism of algebras $\cR_\fs^\op\simeq \cR_{\fs^\vee}$.

Since $D_h^2\simeq\Id$ (see <ref>) we see that in particular $D_h$ is an equivalence of categories (contravariant).
Hence it induces an anti-isomorphism of algebras:

\[\cR_\fs = \Hom_\wG(\Pi_\fs,\Pi_\fs) \simeq \Hom_\wG(D_h\Pi_\fs,D_h\Pi_\fs) = \cR_{\fs^\vee}\]

where in the last equality we used <ref>.

The above corollary means that we have an equivalence of categories $\cR_\fs\lmod\simeq \rmod\cR_{\fs^\vee}$.
In other words, given a right module $V\in\rmod\cR_\fs$  its dual $V^*=\Hom_\bC(V,\bC)$ is naturally a left $\cR_\fs$-module, hence we can view it as a right $\cR_{\fs^\vee}$-module.
Put shortly, taking the dual vector spaces gives us a functor

\[ (-)^*\colon \rmod\cR_\fs\to \rmod\cR_{\fs^\vee}.\]

We are now ready to describe the contragredient on the $\cR_\fs$-module side.
The answer is not surprising at all:

There is a commutative square of functors

\[ 	\begin{tikzcd}
\cM(\wG)_{\fs} \ar[r,"\sim"] \ar[d,"(-)^\vee"'] & \rmod \cR_{\fs} \arrow[d,"(-)^*"]\\
(\cM(\wG)_{\fs^\vee})^\op \ar[r,"\sim"] & (\rmod \cR_{\fs^\vee})^\op.
\end{tikzcd}\]

The fact that the vertical right arrow makes sense was discussed above as a consequence of <ref>

Note that $\Pi_{\fs}$ being projective and finitely generated, together with the calculation from <ref>, implies that we have  natural isomorphisms of functors

\[ \Hom_{\wG}(\Pi_{\fs},-)\simeq \Hom_\wG(\Pi_\fs,\cH(\wG))\otimes_{\cH(\wG)}- \simeq \Pi_{\fs^\vee}\otimes_{\cH(\wG)} - .\]

Let $V\in\cM(\wG)_\fs$ and use tensor-hom adjunction to obtain natural isomorphisms

\begin{align*}
\Hom_\wG(\Pi_{\fs}, V)^* & \simeq \Hom(\Pi_{\fs^\vee}\otimes_{\cH(\wG)} V, \bC)\\
& \simeq \Hom_\wG(\Pi_{\fs^\vee},V^*)\\
& \simeq \Hom_\wG(\Pi_{\fs^\vee},V^\vee)
\end{align*}

where the last equality follows because the image of a smooth module is smooth.

As a final computation, we show that the homological duality on the $\cM(\wG)_\fs$ side goes to homological duality on the $\cR_\fs$-module side:

There is a commutative square of functors

\[ 	\begin{tikzcd}
\cD^b(\cM(\wG)_{\fs}) \ar[r,"\sim"] \ar[d,"D_h"'] & \cD^b(\rmod \cR_{\fs}) \arrow[d,"D_h"]\\
\cD^b(\cM(\wG)_{\fs^\vee})^\op \ar[r,"\sim"] & \cD^b(\rmod \cR_{\fs^\vee})^\op
\end{tikzcd}\]

where the $D_h$ on the right is $\RHom_{\cR_\fs}(-,\cR_\fs)$ and we again use the algebra anti-isomorphism $\cR_\fs \simeq \cR_{\fs^\vee}$ from <ref>.

We deal first with the left-bottom composition.
Since $D_h$ is involutive (see <ref>), it induces an isomorphism on Hom spaces.
In particular, using <ref>, we get an isomorphism of left $\cR_\fs$-modules

\[ \RHom_\wG(\Pi_{\fs^\vee}, D_h(V)) = \RHom_\wG(V,\Pi_\fs)\]

for all $V\in\cD^b(\cM(\wG)_\fs)$.

On the other hand, let us recall that the equivalence $\cM(\wG)_\fs\to \rmod\cR_\fs$ was given by
the functor $\Hom_\wG(\Pi_\fs,-)$.
The tensor-Hom adjunction gives its left adjoint as $-\otimes_{\cR_\fs} \Pi_\fs$.
Being an equivalence implies the adjunction is a bi-adjunction, i.e.,  we also have that $\Hom_{\wG}(\Pi_\fs,-)$ is left adjoint to $-\otimes_{\cR_\fs} \Pi_\fs$.

We can now compute the right-top composition and conclude the proof:

\begin{align*}
\RHom_{\cR_\fs}(\RHom_\wG(\Pi_\fs,V),\cR_{\fs})  &\simeq \RHom_\wG(V,\cR_{\fs}\otimes_{\cR_\fs} \Pi_\fs)\\
&\simeq \RHom_\wG(V,\Pi_\fs),
\end{align*}

natural isomorphisms of left $\cR_\fs$-modules.

Using a similar argument and Morita theory (see for example <cit.>) one shows that
under an equivalence of module categories $R\lmod \simeq
S\lmod$, the $R$-bimodule $R$ is sent to the $S$-bimodule $S$, and as a consequence, the  homological duality for $R$ corresponds to the homological duality for $S$.

As a consequence of this section, in order to study homological duality on $\cM(\wG)$ we can study it on $\rmod \cR_\fs$ for all the cuspidal data $\fs$.
This will play an important role in <ref> where we show that homological duality on finite length cuspidals is isomorphic to the contragredient (up to a shift).

§ THE DUALITY THEOREM OF SCHNEIDER–STUHLER

We will use the duality theorem from <ref>, applied to the Hecke algebra $\cH(\wG)$,
to deduce a theorem of Schneider and Stuhler  <cit.>.
In loc.cit., the result is stated and proved in the subcategory of modules with central character but this restriction was removed in [25].
Our proof is independent both of [31] and of [25], and it, moreover, provides a generalization of the main result of [25] from irreducible modules to modules of finite length.
The approach that we take was suggested in <cit.>.

Let $\fs=[\rho,\wL]\in\cB(\wG)$ be a cuspidal datum and put $d=d(\fs)$ the split rank of the center of $\wL$.
We denote the contragredient cuspidal datum by $\fs^\vee = [\rho^\vee,\wL]$ for which we have $d(\fs^\vee)=d$.

Recall that in <ref> it was proved that, for $\pi\in\cM(\wG)_\fs^\finl$, we have $H^i(D_h(\pi))\neq0$ only for $i=d$.
We put $\bD_h(\pi) = H^d(D_h(\pi))$.

Let $\pi\in\cM(\wG)_\fs^\finl$ be a finite length representation in the Bernstein block $\fs$ and let $\pi'\in\cM(\wG)$ be any smooth representation.
Then the natural pairing

\[ \Ext^{i}_\wG(\pi, \pi')  \times \Ext^{d-i}_\wG(\pi', \bD_h(\pi)^\vee)  \rightarrow
\Ext^{d}_\wG(\pi, D_h(\pi)^\vee)   \to \bC\]

provides an isomorphism

\[ \Ext^{i}_\wG(\pi, \pi')^*\simeq  \Ext^{d-i}_\wG(\pi', \bD_h(\pi)^\vee).\]

If $\pi$ is, moreover, irreducible, then $\Ext^d_\wG(\pi,\bD_h(\pi)^\vee)\simeq \bC$.

The first part is just a reformulation of <ref>.
The second part comes from <ref> by noticing that $D_\Nak(\pi) = \bD_h(\pi)^\vee[d]$.

Note that $(\cdot)^\vee\circ \bD_h$ is an endofunctor of the category $\cM(\wG)_\fs^\finl$ of finite length modules.
We would like to think that $\bD_h$ commutes with the contragredient on finite length modules but we are not able to prove it.
Another way to reformulate it is to say that $(-)^\vee\circ \bD_h$ is an involution on $\cM(\wG)_\fs^\finl$.
<ref> confirms this for finite length cuspidal representations.
More generally, $D_h$ should commute with Grothendieck–Serre duality (over the Bernstein center) for all finitely generated $\wG$-modules.[This was suggested to us by Roman Bezrukavnikov.]
This will be investigated in future work.

§ DUALITIES ON FINITE LENGTH REPRESENTATIONS

In this section we  prove that the homological duality restricted to finite length cuspidal
representations is nothing else but contragredient duality up to a shift
(see <ref>). This is due to the fact that the condition (FsG) in Definition <ref>  is satisfied on a cuspidal block (see <ref>).
Along the way we will show the well-known fact that Grothendieck–Serre duality over the Bernstein center, restricted to finite length modules, is just the contragredient duality (see <ref>).
We make use of the generalities developed in <ref>.

Recall the Bernstein decomposition (<ref>):
the category $\cM(\wG)$ of smooth representations of $\wG$
decomposes into
blocks $\cM(\wG)\simeq \prod_\fs \cM(\wG)_\fs$,
and for each block,  we have $\cM(\wG)_\fs \simeq \rmod \cR_\fs$
(see <ref>).

Moreover, we have shown (see <ref> and <ref>) that under the equivalence $\cM(\wG)_\fs \simeq \rmod \cR_\fs$,
the contragredient on $\cM(\wG)_\fs$ corresponds to taking dual vector spaces on $\rmod\cR_\fs$ and that homological duality on $\cM(\wG)_\fs$ corresponds to homological duality on $\rmod\cR_\fs$.

Recall from  <ref> that the center of the block $\cM(\wG)_\fs$ is $\cZ_\fs:=\cZ(\cR_\fs)$ and that $\cR_\fs$ is finite over $\cZ_\fs$.
From the block decomposition, the (Bernstein) center of the category $\cM(\wG)$ satisfies

\[ \cZ = \prod_\fs \cZ_\fs.\]

Moreover, we know that each $\cZ_\fs$ is the ring of invariants of a finite group acting on a Laurent polynomial ring (see <ref>), therefore it is a Cohen-Macaulay ring, and has a dualizing complex $\omega_{\cZ_\fs}^\circ$.
The dualizing complex for the ring $\cZ$ is then

\[\omega_\cZ^\circ = \prod_\fs \omega_{\cZ_\fs}^\circ.\]

The following is the relative version of <ref> and the boundedness of the derived categories is justified by the fact that $\omega_\cZ^\circ$ has finite injective dimension (see <ref>):

The Grothendieck–Serre duality relative to the center is defined by

\[ D_{GS/\cZ}
=\RHom_\cZ(-,\omega^\circ_\cZ)^\sm\colon \cD^b_{fg}(\cM(\wG))\to \cD^b_{fg}(\cM(\wG))^\op \]

where the superscript “sm”  signifies taking smooth vectors.

The functor $D_{GS/\cZ}$ restricted to admissible modules is isomorphic to the contragredient functor.
In particular, this holds for all finite length $\wG$-modules.

Let $V$ be an admissible module for $\wG$.
For any open compact subgroup $\wK \subset \wG$,  $V^\wK$ is a finite dimensional module over the $\wK$-biinvariant Hecke algebra $\cH(\wG\git \wK)$  and it is also a module over the center $\cZ$.

We have the following natural isomorphisms:

\begin{align*}
\RHom_\cZ(V,\omega_Z^\circ) & = \RHom_\cZ(\bigcup_\wK V^\wK, \omega_Z^\circ)&\\
& = \varprojlim_\wK \RHom_\cZ(V^\wK,\omega_Z^\circ) &\\
& = \varprojlim_\wK (V^\wK)^*& \text{ by \cref{C:D_GS/A on finite length R-mod}} \\
& = V^*,&
\end{align*}

and we conclude by taking smooth vectors.

Grothendieck–Serre duality relative to the center behaves well under equivalences of categories.
We record the following for completeness as an analogue of <ref>	and <ref>.

Let $\fs=[\wL,\rho]\in\cB(\wG)$ be a cuspidal datum.
Then, under the equivalence $\cM(\wG)_\fs\simeq \rmod \cR_\fs$, we have the following commutative square of functors

\[ 	\begin{tikzcd}
\cD^b(\cM(\wG)_{\fs}) \ar[r,"\sim"] \ar[d,"D_{GS}"'] & \cD^b(\rmod \cR_{\fs}) \arrow[d,"D_{GS}"]\\
\cD^b(\cM(\wG)_{\fs^\vee})^\op \ar[r,"\sim"] & \cD^b(\rmod \cR_{\fs^\vee})^\op.
\end{tikzcd}\]

Under an equivalence of categories, the centers are isomorphic.
In this proof we put $Z:=\cZ_\fs$.
So the two $D_{GS}$ functors in the diagram are both $\RHom_Z(-,\omega_{Z}^\circ)$.

<ref> ensures that the target of the left vertical arrow is correct.
Similarly, use <ref> together with <ref> for the right vertical arrow.

We have to show that, for $V\in\cM(\wG)$, there are natural isomorphisms

\begin{align}\label{Eq:D_GS commutes with equiv}
\RHom_Z(\Hom_\wG(\Pi_\fs,V),\omega_Z^\circ) \simeq \Hom_\wG (\Pi_{\fs^\vee},\RHom_Z(V,\omega_Z^\circ)).
\end{align}

Notice that $\Pi_\fs$ is projective, so we don't need to derive the Hom space from it.

Using <ref> and <ref> we have $\Hom_\wG(\Pi_\fs,V)\simeq \Pi_{\fs^\vee}\otimes_{\cH(\wG)} V$.
We can rewrite the left hand-side of (<ref>) and use tensor-hom adjunction to conclude.

Let $\rho\in\cM(\wG)$ be an irreducible cuspidal representation and put as usual $d=\rank_\bZ(\wG/\wGo)$.
It has already been proved that homological duality restricts to a contravariant functor

\[ \bD_h:=[d]\circ D_h\colon \cM(\wG)_{[\rho]}^\finl \to (\cM(\wG)_{[\rho^\vee]}^\finl)^\op,\]

see <ref> and <ref>.
We can see that it is not new:

For $\rho\in\cM(\wG)$ an irreducible cuspidal representation of $\wG$, and  $d=\rank_\bZ(\wG/\wGo)$,
the above functor $\bD_h$ is isomorphic to the contragredient.

The block $\cM(\wG)_{[\rho]}$ is equivalent to the category $\rmod\cR_{[\rho]}$ and this equivalence commutes with homological duality (see <ref>) as well as with contragredient (see <ref>).
Moreover, it sends finite length $\wG$-modules to finite length $\cR_{[\rho]}$-modules.
In order to prove the claim it is therefore enough to do it for $\rmod\cR_{[\rho]}$.

The center of $\cR_{[\rho]}$ is a Laurent polynomial algebra of dimension $d$ (see <ref>) and moreover $\cR_{[\rho]}$ satisfies the condition (<ref>) (a particular case of
the condition (FsG) in Definition <ref>) by <ref>.
Applying <ref> we conclude.

The same argument gives the following corollary.
Let $\rho\in\cM(\wG)$ be an irreducible cuspidal representation and $d=\rank_\bZ(\wG/\wGo)$.
Then on the cuspidal block $\cD^b_{fg}(\cM(\wG)_{[\rho]})$, we have a natural isomorphism of functors

\[ [d]\circ D_h\simeq D_{GS/\cZ}.\]

If $G$ is a torus  then all smooth representations of a finite central extension $\wG$ are cuspidal and therefore the above corollaries apply  to all finite length smooth $\wG$-modules.
Moreover, in this situation the Grothendieck–Serre duality and homological duality agree (up to shift) on all smooth modules.

More generally, the following gives a necessary and sufficient condition for homological duality to be isomorphic to Grothendieck–Serre duality:

On a block $\cD^b(\cM(\wG)_\fs)$, we have that $[d(\fs)]\circ D_h\simeq  D_{GS/\cZ}$ if and only if $\cR_\fs$ satisfies condition (FsG) in  Definition <ref>.

Given <ref>, this is simply a restatement of <ref>.

Let $\fs=[\wL,\rho]$ be a cuspidal datum such that  the stabilizer of the inertia class $[\rho]$ in the relative Weyl group is trivial, i.e., there is no non-trivial $w\in W(\wL,\wG)$ such 	that ${}^w\rho \simeq \rho\chi$ for some unramified character $\chi$ of $L$.
Then by a result of Roche <cit.>, parabolic induction induces an equivalence of blocks $\cM(L)_{[\rho]}\to\cM(G)_{\fs}$ and therefore a Morita equivalence of  $\cR_{[\rho]}$ and $\cR_{\fs}$.
As $\cR_{[\rho]}$ is Azumaya by <ref>, the same is true of $\cR_\fs$.
By the general property of Azumaya algebras recalled in <ref>, we find that the algebra $\cR_\fs$ satisfies condition

By <ref>,
the Grothendieck–Serre duality on irreducible representations of $\wG$
is simply the contragredient. On the other hand, as it is well known, the homological duality
(same as Aubert-Zelevinsky duality for irreducible representations) interchanges the trivial representation and the Steinberg representation
of $\SL_2(F)$. Therefore the Corollary above is not true for all representations of $\SL_2(F)$ which means that
the        condition (FsG) in Definition <ref> does not hold for the Iwahori block for $\SL_2(F)$.

Let us also note the particular case of a free abelian group $\Gamma\simeq \bZ^d$. (The
abelian category of representations of $\Gamma$ is the Iwahori block of
a split torus over any $p$-adic field, so in considering this example, we
are not going outside the realm of $p$-adic groups!)
The category of representations of $\Gamma$
is the same as  the category of representations of the group algebra $\bC[\Gamma]\simeq \bC[X_1^\pm,\dots,X_d^\pm]$.
From <ref> we get:

The Grothendieck–Serre duality, (shifted by $d$) homological duality and contragredient duality all agree on finite length $\Gamma$-modules:

\[ D_{GS} = [d]\circ D_h = (-)^* \text{ as functors on } \cM(\Gamma)^\finl.\]

Equivalently, for a finite length module $M$ over $A= \bC[X_1^\pm,\dots,X_d^\pm]$,

\[ \Ext_A^d(M, A) \cong \Hom_\bC(M,\bC),\]

an isomorphism of $A$-modules,                an assertion which can also be made for  a finite length module $M$ over $A= \bC[X_1,\dots,X_d]$.

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DF: IRMA Institut de Mathematique
7 rue René-Descartes, 67000 Strasbourg

Email<EMAIL_ADDRESS>
DP:  Indian Institute of Technology Bombay, Mumbai.

Email<EMAIL_ADDRESS>
We have the following natural isomorphisms:

\begin{align*}
\Hom_\wG(\bfi_{\wL,\wP}^\wG(V),W) &
= \Hom_\wG(\ind_\wP^\wG \Res_\wP^\wL (\delta_P^{1/2}\otimes V),W)  & \text{ by definition}\\
&=\Hom_\wG(\cH(\wG)\otimes_{\cH(\wP)}\Res_\wP^\wL(\delta_P^{-1/2}\otimes V),W) & \text{by \cref{P:ind_P^G as tensor product}}\\
& = \Hom_\wP(\Res_\wP^\wL(\delta_P^{-1/2}\otimes V),\overline{W}) & \text{by \cref{P:right adjoint to ind_P^G}}\\
& = \Hom_\wL(\delta_P^{-1/2}\otimes V,\overline{W}^N) & \Res_\wP^\wL \dashv (-)^N\\
& = \Hom_\wL(V,\delta_P^{1/2}\otimes \overline{W_{N^-}})&\text{by \cref{C:invar and coinvar are isomorphic}}\\
& = \Hom_\wL(V,\delta_P^{1/2} \otimes W_{N^-})& \text{ by \cref{R:non-deg is right adjoint}}\\
& = \Hom_\wL(V,\bfr_{\wL,\wP^-}^\wG(W)) & \text{ by definition.}
\end{align*}

MISSING 2


# Unleashing the Power of Graph Learning through LLM-based Autonomous Agents

Lanning Wei1,2,3, Zhiqiang He1,4, Huan Zhao3,∗, Quanming Yao5,∗ 1Institute of
Computing Technology, Chinese Academy of Sciences 2University of Chinese
Academy of Sciences 34Paradigm. Inc.,4Lenovo, 5Department of Electronic
Engineering, Tsinghua UniversityBeijingChina<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Graph structured data are widely existed and applied in the real-world
applications, while it is a challenge to handling these diverse data and
learning tasks on graph in an efficient manner. When facing the complicated
graph learning tasks, experts have designed diverse Graph Neural Networks
(GNNs) in recent years. They have also implemented AutoML in Graph, also known
as AutoGraph, to automatically generate data-specific solutions. Despite their
success, they encounter limitations in (1) managing diverse learning tasks at
various levels, (2) dealing with different procedures in graph learning beyond
architecture design, and (3) the huge requirements on the prior knowledge when
using AutoGraph. In this paper, we propose to use Large Language Models (LLMs)
as autonomous agents to simplify the learning process on diverse real-world
graphs. Specifically, in response to a user request which may contain varying
data and learning targets at the node, edge, or graph levels, the complex
graph learning task is decomposed into three components following the agent
planning, namely, detecting the learning intent, configuring solutions based
on AutoGraph, and generating a response. The AutoGraph agents manage crucial
procedures in automated graph learning, including data-processing, AutoML
configuration, searching architectures, and hyper-parameter fine-tuning. With
these agents, those components are processed by decomposing and completing
step by step, thereby generating an solution for the given data automatically,
regardless of the learning task on node or graph. The proposed method is
dubbed Auto2Graph, and the comparable performance on different datasets and
learning tasks. Its effectiveness is demonstrated by its comparable
performance on different datasets and learning tasks, as well as the human-
like decisions made by the agents. 111Lanning is a research intern in
4Paradigm. $\ast$: corresponding author.

##  1\. Introduction

Graph-structured data have been widely employed in various real-world domains,
such as chemistry and biomedical molecules (Zhang et al., 2023c; Gilmer et
al., 2017), social networks (Hamilton et al., 2017), e-commerce graphs (Lü et
al., 2012), and more. These applications, rooted in graphs, exhibit
substantial diversity, necessitating distinct methods for effective graph
mining (You et al., 2020). In recent years, Graph Neural Networks (GNNs) have
gained widespread traction owing to their promising performance on graph-
structured data (Kipf and Welling, 2016). Numerous GNN variants have been
crafted to cater to different applications, as exemplified by works such as
(Kipf and Welling, 2016; Hamilton et al., 2017; Xu et al., 2019, 2018; Ying et
al., 2018; Zhang and Chen, 2018). The task of devising appropriate solutions
for diverse data becomes both an imperative and a challenge, demanding an
efficient and effective approach (Gao et al., 2020; Zhao et al., 2020; Wei et
al., 2021; Wang et al., 2022b; Zhang et al., 2022b).

AutoML’s potential for automatically generating data-specific solutions for
graphs is promising (Wang et al., 2022a). However, it has limited scope in
machine learning tasks, and requires expertise, which can be a challenge for
non-expert users. Firstly, AutoML cannot handle all procedures in machine
learning tasks. To be specific, AutoML is proposed to achieve better
performance with less human assistance and high computational efficiency (Zoph
and Le, 2017; Pham et al., 2017, 2018; Yao et al., 2018; He et al., 2021). Its
utility extends to autonomous feature engineering, Neural Architecture Search
(NAS) (Gao et al., 2020; You et al., 2020; Qin et al., 2022), Hyper-Parameter
Optimization (HPO) (Tu et al., 2019; Wang et al., 2021a; Luo et al., 2019),
spanning various machine learning tasks, and extending to an array of graph-
based learning endeavors (Zhang et al., 2021; Wang et al., 2022a; Gao et al.,
2020; Wang et al., 2022b; Wei et al., 2021; Ding et al., 2021). However, it
lacks the capability to automatically identify tasks and choose suitable
solutions, nor can it generate responses for given tasks. This implies that
humans must design and select task-specific solutions accordingly. Second, the
significant requirement for prior knowledge presents the biggest challenge in
utilizing AutoML. Despite its success in less human assistance, the effective
use of AutoML heavily relies on extensive prior knowledge in the field and
necessary code support. As a result, non-expert users may face difficulties
when trying to apply AutoML for general machine learning tasks (Tornede et
al., 2023).

This paper endeavors to bridge the gap between a simplified approach to
learning graph-structured data and the diverse data and tasks encountered in
real-world, and we leverage Large Language Models (LLMs) as autonomous agents
to achieve this. Recently, LLMs have undergone a rapid succession of
breakthroughs, endowing them with a formidable capacity to emulate human-like
decision-making processes (Wang et al., 2023; Yuan et al., 2023). They can
empower the machine learning tasks by automatically decomposing and completing
the tasks directly (Shen et al., 2023; Zhang et al., 2023b; Wang et al., 2023;
Zhang et al., 2023a; Yuan et al., 2023), enhancing the performance from data
understanding and architecture design, (Guo et al., 2023; Chen et al., 2023),
and even directly predict the prediction (Zhang, 2023; Gao et al., 2023).
Considering the full machine learning procedures and the insufficient ability
of LLMs in understanding the graph-structured data (Guo et al., 2023; Chen et
al., 2023), in this paper, we adopt the LLM-based autonomous agents to empower
the complex procedures in graph learning, and automatically provide more
convenient and comprehensive solutions when dealing with different tasks and
data. Additionally, we incorporate AutoML to further increase efficiency and
effectiveness in these procedures. The method designed in this study is
referred to as Auto2Graph.

In graph learning tasks, the procedures are fixed and ordered as shown in Fig.
1. This framework encompasses distinct phases such as intent detection,
AutoGraph agents, which collectively facilitate the transition from user
queries to final outputs. Especially, four AutoGraph agents are designed
considering each key procedure in automated graph learning (He et al., 2021).
They are responsible for data-processing, configuration AutoML, searching
architectures and fine-tuning hyper-parameters, respectively. Based on the
designed framework, the different data and tasks provided in the user request
are detected as different intents on graph learning. Then, all agents
decompose the complex tasks into different steps. Subsequently, they complete
these tasks either with their strong reasoning ability or external knowledge
and tools, with which the AutoGraph agents could generate data and task
specific settings accordingly. Once learning is completed, responses are
generated based on the learning process and user request. As a result,
different data and tasks can be solved in different approaches, and the
learning process can be accelerated with the AutoML. Then, the feasibility of
LLM-based autonomous agents for applying to diverse data and learning tasks
are obvious. Furthermore, in the experiments, we provide an instance to
evaluate the designed method, in which the agents can make correct decisions
following the designed prompts.

The main insights we want to bring across with this work are the following:

  * •

We propose a method Auto2Graph to autonomously solving the different graph-
structured data and learning tasks, and it is achieved based on LLMs with
their the strong ability in decomposing and completing the complicated tasks.

  * •

A graph learning framework based on LLM is proposed, which contains diverse
LLM agents, i.e., intent detection, four AutoGraph agents and response agent,
to assist the diverse non-expert users when handling their graph-structured
data.

  * •

We provide an instance to evaluate the effectiveness of proposed Auto2Graph.
The results from both agents and experiments demonstrate the feasibility and
effectiveness of the designed method.

Figure 1. The designed framework when learning on graph data with the help of
LLM agents.

##  2\. Related work

###  2.1. LLM-empowered machine learning

LLM have achieved great success in wide applications with their ability in
achieving human-like decisions. They are used to serve as the excellent
artificial general intelligence (AGI) in real world, with applications
including question answering in natural language processing tasks, automatic
solving compute vision tasks, and reasoning graph properties (Guo et al.,
2023; Wang et al., 2023; Zhang, 2023; Shen et al., 2023).

On the one hand, LLM could empower the machine learning tasks by automatically
decomposing and completing complex ML tasks, and they are proposed to bringing
more convenient, comprehensive and reliable decisions when facing diverse
applications and tasks (Shen et al., 2023; Zhang et al., 2023b; Wang et al.,
2023; Zhang et al., 2023a; Yuan et al., 2023). For instance, huggingGPT (Shen
et al., 2023) responds to the user request with task plan, model selection,
model execution and response generation procedures. Each of them are executed
automatically following the managing of LLM-based agents. Similarly, AutoML-
GPT (Zhang et al., 2023b) tackles diverse datasets and tasks automatically via
conducting experiments from data processing to model architecture, hyper-
parameter tuning. These procedures are also managed by LLMs. On the other
hand, LLM can enhance machine learning tasks from a model perspective. This
includes, but is not limited to, make a better under standing on data (Guo et
al., 2023; Zhang, 2023; Chen et al., 2023), suggest neural architectures (Yu
et al., 2023; Zheng et al., 2023; Zhang et al., 2023b), or directly predict
labels with LLMs (Zhang, 2023; Sun et al., 2023; Gao et al., 2023; Chen et
al., 2023).

In this paper, we aim to unleash the potential of LLM in graph learning
considering the diverse data and tasks. Given the limitation of LLMs in
understanding the graph-structured data as highlighted in (Chen et al., 2023;
Guo et al., 2023), we have opted not to propose an architecture directly
derived from LLM for these tasks. Instead, we are focusing on using LLM
autonomous agents to manage complex procedures when dealing with a variety of
graph-structured data and tasks.

###  2.2. Automatic Graph Representation Learning

In the literature, most AutoGraph methods focus on designing architectures,
while others concentrate on feature engineering and hyper-parameter
optimization.

Neural architecture search (NAS) methods were proposed to automatically find
SOTA CNN architectures, and they require a search space to represent the
architectures and a search algorithm to explore the search space (Zoph and Le,
2017; Real et al., 2017; Pham et al., 2018). Recent years, researchers employ
NAS in designing GNNs (Wang et al., 2022a) which can largely alleviate the
human labor in selecting effective GNNs. GraphNAS (Gao et al., 2020), AutoGM
(Yoon et al., 2020) and DSS (Li et al., 2021) focus on designing aggregation
operations in each layer. Apart from this, NAS-GCN (Jiang and Balaprakash,
2020) learns adaptive global pooling functions additionally, and PAS (Wei et
al., 2021) is proposed to design pooling operations for the graph
classification task. Beyond GNN layer, SANE (Zhao et al., 2021), AutoGraph (Li
and King, 2020) and F2GNN (Wei et al., 2022) further designs the skip-
connections in GNNs. DFG-NAS (Zhang et al., 2022a) and PASCA (Zhang et al.,
2022b) separate the message construction and representation update, which
allows for deep GNNs and enables to applied on large-scale datasets. GraphGym
(You et al., 2020) provides the design dimension of inter-layer, intra-layer
and training strategy dimensions when design GNNs for node and graph
classification task. For feature engineering on graph structured data,
AutoField (Wang et al., 2022c) and AutoCross (Luo et al., 2019) aim to
automatically selecting feature subsets for recommendation system. MDE (Ginart
et al., 2021), AutoEmb (Zhaok et al., 2021) and ESAPN (Liu et al., 2020) focus
on designing the feature dimensions given the unappealing performance caused
by the long-tailed distributed features. Besides, AutoNE (Tu et al., 2019) and
e-AutoGR (Wang et al., 2021b) are proposed to optimize the hyper-parameters on
the condition of large-scale graphs.

In this paper, we introduce four agents to manage the procedures in AutoGraph,
namely, data-processing, AutoML configuration, architecture search , and
hyper-parameter fine-tuning. The aforementioned methods focus on improving one
of the procedure merely. Therefore, the proposed method is orthogonal to these
methods, and they can be employed in this paper to further enhance efficiency
and effectiveness when learning on graphs.

##  3\. Method

As shown in Fig. 1, general machine learning solutions are designed following
the procedures of data pre-processing, model selection, model optimization and
evaluations, and AutoML assists these procedures by automatically processing
data, searching and optimizing neural architectures (He et al., 2021). To
further alleviate the complexities of the learning procedures and reduce the
need for prior knowledge in designing data-specific solutions with AutoML, we
utilize LLM. This assists users in tasks such as intent detection, planning
and completing graph learning tasks, model execution, and response generation
when learning on graph-structured data.

To be specific, when learning on graph-structured data, the user describes the
data and learning target using natural language instead of configuring and
writing code. Firstly, the LLM-empowered agents detect the learning intent
from the given request. Then, several agents are arranged to automatically
learn on graphs in a step-by-step process, from data processing to
architecture optimization. Finally, responses are automatically generated for
the user based on the search results. By implementing this method, various
users can automatically learn from diverse graphs with the help of LLM-based
agents, without needing extensive knowledge of AutoML or coding skills. The
subsequent sections provide detailed insights into how LLM agents alleviate
the labor costs associated with code generation and enhance the utilization of
user-specific prior knowledge.

###  3.1. Preliminary

The designed procedures necessitate agents to comprehend requests, invoke
APIs, utilize web retrieval, and write code.. Given the complexity of these
tasks, we utilize the LangChain 222https://github.com/langchain-ai/langchain,
an LLM-driven framework, to interact with external resources and models, to
assist in different steps of designing GNNs. Additionally, we use PyG
333https://github.com/pyg-team/pytorch_geometric for designing GNNs, and the
interactions with this package are achieved using LLM.

Figure 2. The introduction of procedures and agents used when learning on
graphs.

###  3.2. Intent Detection

The process begins with the preparation of data and the learning target. An
agent is proposed to identify the graph learning intents and extract prior
user-preferred knowledge or constraints based on LLMs. The user’s request may
encompass various elements including: {‘data’ $\mathcal{D}$: the introduction
of graph or the used dataset. ‘task’ $\mathcal{T}$: the learning target on the
graph., ‘Evaluation metric’ $\mathcal{M}$: the evaluations of the searched
architectures., ‘Preference’ $\mathcal{P}$: the prior knowledge in designing
and training the architectures on data., ‘Constrains’ $\mathcal{C}$: the
constrains on model, device, or training strategy. }. These pieces of
information may be organized in various ways by different users, and the agent
is responsible for sequentially extracting the relevant information from the
user’s request, which is then utilized in the subsequent procedures. By
leveraging LLM, we automate the interactions between users and the learning
task, eliminating the need for users to configure the code. This streamlined
approach significantly reduces the user’s workload and enhances the overall
efficiency when facing diverse data and learning targets.

###  3.3. Agent organization to empower AutoGraph

Based on the extracted intents from user request, we provide several agents
which co-operated to design data-specific solutions automatically.

Data agent. The data agent is designed to prepare the data for the following
searching and tuning procedures. Depending on the learning task and evaluation
metric, the agent’s objective is to select the appropriate function for data
loading, feature processing, and data splits. By exploring the PyG documents,
these procedures can be accomplished by invoking the correct functions.

Configuration agent. When searching data-specific GNNs based on NAS, we first
need to set the configurations of the Graph NAS, i.e., the search space,
search algorithm and evaluation metric.

$\bullet$Search Space. The search space is constructed by preparing a set of
candidate operations within the designed architecture backbone, in which the
agent plays a crucial role in make decisions. Firstly, the agent selects the
operation modules based on the learning task $\mathcal{T}$. Then, to prepare
the potential operations for the intended dimension, the agent refers to the
corresponding documents in PyG, from which the available candidate operations
can be extracted. Considering the different preference $\mathcal{P}$ and
constrains $\mathcal{C}$ which may affect the model design, these candidate
operations are adjusted correspondingly by agents. By leveraging user feedback
and preferences, the configuration agent assists in tailoring the design space
to better align with the specific requirements and objectives of the users.

$\bullet$Search algorithm. The search algorithm is proposed to explore the
designed search space towards high-performing candidate architectures. The
configuration agent assists in selecting the appropriate algorithm based on
user requirements, taking into account constraints such as time, memory, and
the number of parameters. Empirically speaking, the feasibility of search
algorithm is closely tied to the characteristics of the search space and the
desired efficiency. Generally, differentiable search algorithms exhibit high
efficiency due to their parameter sharing strategy and continuous relaxation
(Liu et al., 2019; Xie et al., 2018). However, certain operations, such as the
pooling operation (Wei et al., 2021), pose challenges for direct continuous
relaxation due to the mismatched node problem. To address this, relaxation
strategies or policy gradient approaches can be employed. The agent rely on
LLM to select the appropriate algorithm and based on the specific requirements
and preferences expressed by the user.

Searching agent. Based on the configurations on the search space and
algorithm, we construct the models and then search GNNs, in which the
searching agent helps to configure the hyper-arguments and push the process
forward. To be specific, the execution of the code is facilitated by
constructing the hyper-arguments required for the model. At this stage, the
searching agent generates and executes the code to search GNNs. Once the
search is complete, the agent extracts the searched GNNs by referencing the
log file. Finally, it is responsible for formatting the output. Through this
collaborative process, we obtain the searched architectures that fulfill the
specified design objectives and requirements.

Tuning agent. After obtaining the searched GNNs, we can fine-tune the hyper-
parameters, such as learning rate, dropout ratio, weight decay ratio, and
normalization method, to obtain the final results. For simplicity, we utilize
HyperOPT (Bergstra et al., 2013) to accomplish this. The tuning agent assists
in selecting appropriate inputs based on the searched GNNs, learning task, and
preference. Subsequently, the agent aids in executing the code and selecting
performance metrics from the log file.

###  3.4. Response generation

We have obtained the searched GNNs and the corresponding results, and then
utilize a response agent to organize these results into a comprehensive
paragraph. The response agent bases its organization on, but is not limited
to, the following:  {‘Searched GNNs’ $\bm{\alpha}$: the searched
architectures., ‘Prediction results’ $\mathcal{R}$: the required results
mentioned in the user request., ‘Hyper-parameters’ $\mathcal{H}$: the searched
hyper-parameters., ‘Resource consuming’: the time and memory cost. } . It is
easier for response agent due to its power in processing natural language.

###  3.5. Discussion

Fewer requirement on the expert knowledge. Typically, experts design neural
architectures through a trial-and-error process. However, this approach may
fail to explore potentially superior architectures due to the limited range of
candidate designs. AutoML expands the exploration field by designing a search
space that can encompass millions of candidates. Nevertheless, the design of
the search space and search algorithms must be carefully considered for
efficiency and effectiveness, as they significantly impact the performance of
the searched architectures (You et al., 2020; Gao and Ji, 2019; Wei et al.,
2022). Essentially, expert knowledge is necessary to comprehend and configure
AutoML for use with private data. In this paper, we aim to connect diverse
data with their corresponding solutions through the use of LLM autonomous
agents. Users provide descriptions of the learning target, after which the LLM
agents decompose and complete the complicated learning task step by step. This
process is facilitated by the LLM’s reasoning ability and the designed
framework, which supports the integration of prior knowledge.

Flexible extension to wide spectrum graph data and tasks. In the literature,
most AutoGraph methods mentioned in Section 2.2 are designed for fixed graph
types which handle single task and only use fixed search space instead. When
confronted with graphs that deviate from outside scope, any existing method
with fixed target task and search space may not adequately meet their demands
and capture their unique preferences and characteristics. This can lead to the
existing methods hard to applied due to the unsuitable scenarios on the
learning task, data domain and prior knowledge. This paper proposes bridging
this gap using LLM-based agents. These agents interact directly with computers
and APIs using natural languages, allowing for the incorporation and
processing of user expertise, such as data processing and architecture
selection preferences. By capitalizing on user expertise and domain-specific
insights, we can develop a more universal approach that can efficiently handle
a broad spectrum of graph learning scenarios. This allows us to customize and
fine-tune AutoGraph to better match the unique requirements and
characteristics of various graph types and different learning targets, whether
at node, edge, or graph levels.

Further revision with human/experimental feedback. For simplicity, we have
omitted feedback within the designed framework when completing the tasks,
advancing procedures solely based on user requests. However, it is often
challenging to accomplish tasks in one stroke, and both successful and
unsuccessful experiences can enhance the solution development and planning
capabilities of LLMs (Wang et al., 2023). The proposed method could be further
enhanced by integrating feedback from experimental results and users into each
agent. This would result in more reliable and comprehensive decision-making
agents.

##  4\. Experiments

To evaluate the feasibility of the designed method, in the experiment, we
first conduct an implementation instance of AutoGraph which contains a
simplified configurations of the search space and hyper-parameters. Then, we
provide the performance comparisons of this instance and the detailed prompts
used in the agents. The considerable performance and the human-like decisions
demonstrate its effectiveness.

###  4.1. Implementation instance

In this paper, we focus on designing solutions when facing diverse data and
tasks in real-world, and the LLM based agents and AutoML are employed to
achieve this. Here, we provide one instance of AutoML and show how agents are
conducted on them.

Figure 3. The designed architecture backbone. (a) The macro topology design of
GNNs. (b) The micro operation modules employed in the backbone.

Architecture backbone. We begin by presenting the architecture backbone that
are tailored for different tasks. As illustrated in Fig. 3, we construct the
search space form the micro operation module to the macro architecture
topology. In operation module, the aggregation operation is provided to update
node representations H, and it is relevant for all tasks. Besides, the readout
operations and pooling operations are designed for graph classification tasks
as an example, which responsible for generating graph representations z and
coarse graph $G^{\prime}=(\textbf{H}^{\prime},\textbf{A}^{\prime})$,
respectively. In macro-level, we design the topology of GNNs following F2GNN
(Wei et al., 2022) which enhances the model effectiveness and is independent
of the operations used, making it an orthogonal consideration in the design
process. It is achieved by the selection and fusion operation, in which the
former is designed for each aggregation operation to select inputs from
previous operations, and the latter is designed to integrate these inputs as
the inputs.

Table 1. The candidates of architectures and hyper-parameters. | Module | Task | Operations
---|---|---|---
Model | Aggregation | All |  GCN, SAGE
Readout | Graph |  global_sum, global_mean
Selection | All |  ZERO, IDENTITY
Fusion | All |  sum, mean
Pooling | Graph | None
Hyper- parameters | Learning rate | Node | [0.001, 0.005]
Learning rate | Graph | [0.01, 0.05]
Weight decay | Node | [0.0001, 0.0005]
Weight decay | Graph | [0.001, 0.005]
| Dropout rate | All | [0, 0.5]

Configurations on search space and hyper-parameters. Based on these operation
modules, we pre-define a set of basic operations as candidates as shown in
Table 1, and it can be further extend or modified by the agent. As shown in
Fig. 2, the agent configures the proper modules based on the learning task
$\mathcal{T}$, and then revise the operations based on the users’ preferences
through looking up the PyG documents. The configurations of the hyper-
parameters are as well, and the prompt design can be found in Table 2.
Compared with the fixed search space (or pruned space) and targeted task used
in existing AutoGraph methods, the proposed method can configure them based on
the prior knowledge from users’ request automatically, leading to a better
generalization to the real-world data.

Table 2. An example of agents used in the framework. Agent | Procedures |  Prompt input |  Prompt output
---|---|---|---
Intent | Indent detection |  I have a dataset $D$, like NCI1 dataset in graph benchmark, and each graph is a chemical compounds. I want to find one GNN that has better accuracy. |  $\mathcal{D}$: chemical compounds NCI1; $\mathcal{T}$: graph classification; Evaluation metric $\mathcal{M}$: accuracy. $\cdots$
Configuration-space | P1 |  We have a set of operations $\mathcal{A}$ and the given task $\mathcal{T}$, and you job is to select the proper operations set $\mathcal{A}^{\prime}$ based on $\mathcal{A}$.For each key in the set, you need to justify whether ’op’ should be used when we have a graph learning task is $\mathcal{T}$.The response is the list of all the possible op. |  The response list is [‘aggregation’, ‘pooling’, ‘readout’,‘selection’,‘fusion’]
P2&P3 |  I want to find one API from PyG (pytorch geometric).The keywords of this API may incorporating: ‘graph cheb operation’, and its function may be in one of the operation list. |  ’ChebConv’, ’torch_geometric.nn.conv.cheb_conv.ChebConv’
P4 |  I want to merge the space dict $\mathcal{A}^{\prime}$ with the newly added operation ‘ChebConv’. |  {‘aggregation’: [ GCN, SAGE, ChebConv ],‘readout’: $\cdots$}
Algorithm | P1 &P2 |  I have the following descriptions, your job is predict whether to use the ‘Random’ or ‘differentiable search algorithm’ based on the given task $\mathcal{T}$, design dimension: [‘aggregation’, ‘pooling’, ‘readout’,‘selection’,‘fusion’], and specific design space: $\mathcal{A}^{\prime}$.Please check the principles in the following and return the response. (following the constrains mentioned in Section 3.3.) |  You should use“differentiable search algorithm”.
Searching | P1 |  Your job is to the complete the following code based on $\mathcal{D},\mathcal{T}$ and the selected algorithm. ‘–data –learning_task –algo ’ |  ‘–data NCI1 –learning_task graph –algo diff’
P2 |  - |  os.system(”python script.py” + parser parameters
P3&P4 |  Your job is to find the search architecture based on the loggings, which is formatted as ’Final search architecture: ’. |  ’{GCN, SAGE, GCN, $\cdots$}’

###  4.2. Experimental settings

####  4.2.1. Datasets

To evaluate the designed method when applying to diver data, we adopt the node
classification and graph classification tasks with widely-used datasets as
shown in Table. 3 and Table. 4, respectively.

In the node classification task, Cora (Sen et al., 2008) is the citation
network where each node represents a paper, and each edge represents the
citation relation between two papers; Computers and Photo (Shchur et al.,
2018) are the Amazon co-purchase graphs where nodes represent goods that are
linked by an edge if these goods are frequently bought together; Physics
(Shchur et al., 2018) is a co-authorship graph where nodes are authors who are
connected by an edge if they co-author a paper. In the graph classification
task, NCI1 and NCI109 (Wale and Karypis, 2006) are datasets of chemical
compounds; DD and PROTEINS (Dobson and Doig, 2003) are datasets both protein
graphs.

Table 3. The statistics of four node classification datasets. Datasets | #Nodes | #Edges | #Features | #Classes
---|---|---|---|---
Cora (Sen et al., 2008) | 2,708 | 5,278 | 1,433 | 7
Computers (Shchur et al., 2018) | 13,381 | 245,778 | 767 | 10
Photo (Shchur et al., 2018) | 7,487 | 238,162 | 745 | 8
Physics (Shchur et al., 2018) | 34,493 | 495,924 | 8,415 | 5
Table 4. The statistics of four graph classification datasets. Dataset | # Graphs | # Feature | # Classes | Avg.# of Nodes | Avg.# of Edges
---|---|---|---|---|---
NCI1 (Wale and Karypis, 2006) | 4,110 | 89 | 2 | 29.87 | 32.3
NCI109 (Wale and Karypis, 2006) | 4,127 | 38 | 2 | 29.69 | 32.13
DD (Dobson and Doig, 2003) | 1,178 | 89 | 2 | 384.3 | 715.7
PROTEINS (Dobson and Doig, 2003) | 1,113 | 3 | 2 | 39.1 | 72.8

####  4.2.2. Baselines

In this paper, LLM-based agents focus on decompose and complete the tasks,
rather than improve the model expressiveness. Therefore, we evaluate the
effectiveness compared with existing human-designed methods and the AutoGraph
methods.

In the node classification task, we adopt the (1) human-designed method GPR-
GNN (Chien et al., 2021) that learns the weights of each layer based on
generalized PageRank, and ACM-GCN (Luan et al., 2022) that adaptive mixing the
channel information from low-/high-/full-frequency; and the (2) AutoGraph
method SANE (Zhao et al., 2021) that learns the connections based on JKNet (Xu
et al., 2018) and F2GNN (Wei et al., 2022) that designs the network topology
from the feature fusion perspective. In the graph classification task, we
adopt the (1) the human-designed global pooling method DGCNN (Zhang et al.,
2018) and hierarchical pooling methods DiffPool (Ying et al., 2018) to learn
the hierarchy nature in the graphs; and the (2) AutoGraph method LRGNN (Wei et
al., 2023) which design the network topology to capture the long-range
dependency.

###  4.3. The evaluations of the feasibility

In this section, we show the performance comparisons with the baselines to
evaluate the feasibility of the proposed method and the implemented instance.
As shown in Table 5, our method achieves comparable performance across four
datasets, even when compared to carefully designed baselines that focus on
either architecture topology or aggregation operation. The superiority of our
method becomes evident when applied to graph classification datasets as shown
in Table 6, outperforming those baselines mentioned before. These results
underscore the effectiveness of our implementation, which relies solely on
basic aggregation and readout operations. The performance can be further
enhanced by integrating recent aggregation operations with feedback, as
discussed in Section 3.5.

The interactions with agents are provided in Table 2 when evaluating on the
NCI1 dataset. It is observable that the agents can make decisions with
accuracy comparable to humans. This demonstrates the feasibility of using LLM-
based agents to manage the procedures in graph learning.

Table 5. The performance comparisons on the node classification task. | Cora | Photo | Computer | Physics
---|---|---|---|---
Auto2Graph | 86.81(0.40) | 96.40(0.16) | 92.47(0.20) | 96.93(0.98)
SANE | 86.40(0.38) | 94.53(0.22) | 90.25(0.31) | 98.28(0.13)
F2GNN | 87.42(0.42) | 95.38(0.30) | 91.42(0.26) | 96.92(0.06)
GPRGNN | 87.62(0.48) | 91.93(0.26) | 88.90(0.37) | 97.51(0.21)
ACM-GCN | 86.67(0.14) | 94.35(0.65) | 88.58(0.39) | 97.89(0.23)
Table 6. The performance comparisons on the graph classification task. | Proteins | DD | NCI1 | NCI109
---|---|---|---|---
Auto2Graph | 75.38(5.03) | 78.10(3.21) | 82.14(1.74) | 81.25(1.53)
LRGNN | 75.39(4.40) | 78.18(2.02) | 82.51(1.37) | 81.39(1.92)
DGCNN | 73.95(3.04) | 61.63(5.33) | 76.08(1.03) | 74.58(3.99)
DiffPool | 75.11(2.14) | 77.85(3.53) | 75.04(1.98) | 71.48(2.46)

##  5\. Conclusion

In this paper, we propose to address the challenge to design appropriate
solutions for diverse graph structured data and learning tasks with the help
of LLM-based autonomous agents. When facing the complicated graph learning
task described by users, the autonomous agents plan the graph learning
procedures from detecting the learning intent, designing neural networks based
on AutoGraph, to generating response. Four AutoGraph agents are designed to
obtain the data-specific architectures in an efficient manner. By doing this,
diverse data and learning tasks can be addressed automatically in one
framework, with fewer requirement of prior knowledge and limitations on code
ability. The comparable performance of the implemented instance and prompt
details can demonstrate the feasibility of the designed framework. In the
future work, we will revise the framework towards easier to use, comprehensive
and reliable graph learning tools.

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