The Iwahori-Hecke Algebra, the Ramanujan Conjecture, and Expander - - PowerPoint PPT Presentation

The Iwahori-Hecke Algebra, the Ramanujan Conjecture, and Expander Graphs

Cristina Ballantine

College of the Holy Cross

ICERM April 16, 2013

Graph Theory

• Graph X = (V , E)

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

1 2 3 4

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

1 2 3 4

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

1 2 3 4

• adjacency matrix of X: A = aij

aij = 1 if {vi, vj} ∈ E if {vi, vj} ∈ E

Graph Theory

• Graph X = (V , E)
• V = {v1, v2, . . . , vm} set of vertices
• E set of edges (subsets of order 2 of V )

1 2 3 4

• adjacency matrix of X: A = aij

aij = 1 if {vi, vj} ∈ E if {vi, vj} ∈ E     1 1 1 1 1 1 1 1 1 1    

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a closed,

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a closed, backtrack-less,

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a closed, backtrack-less, tail-less,

Primes in Graphs

Graph X = (V , E)

• Path C

C = (v0, v1, v2, . . . , vn), (vi−1, vi) ∈ E, ∀ 1 ≤ i ≤ n length of path: |C| = n

• Prime [C]

equivalence class of a closed, backtrack-less, tail-less, primitive path C

Zeta Function for Graphs

ZX(u) =

• [C] prime
• 1 − u|C|−1

, u ∈ C

Zeta Function for Graphs

ZX(u) =

• [C] prime
• 1 − u|C|−1

, u ∈ C

Theorem (Ihara 1966)

If X = (V , E) is a finite, connected, (q + 1)-regular multigraph (q odd), then ZX(u)−1 = (1 − u2)r−1 det(I − uA + qu2I), where r = |E| − |V | + 1 and I is the |V | × |V | identity matrix.

Riemann Hypothesis for Graphs

Definition

A finite, connected, (q + 1)-regular graph X satisfies the Riemann Hypothesis if for Re(s) ∈ (0, 1)

• ZX(q−s)

−1 = 0 implies Re(s) = 1 2

Riemann Hypothesis for Graphs

Definition

A finite, connected, (q + 1)-regular graph X satisfies the Riemann Hypothesis if for Re(s) ∈ (0, 1)

• ZX(q−s)

−1 = 0 implies Re(s) = 1 2

• Which graphs satisfy the Riemann Hypothesis?

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

• λ(X) = max{|λ(X)| : λ ∈ Spec(X), |λ| = k}

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

• λ(X) = max{|λ(X)| : λ ∈ Spec(X), |λ| = k}
• ∂(F) = {{u, v} ∈ E | u ∈ F, v ∈ V \ F} (boundary of F)

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

• λ(X) = max{|λ(X)| : λ ∈ Spec(X), |λ| = k}
• ∂(F) = {{u, v} ∈ E | u ∈ F, v ∈ V \ F} (boundary of F)

Definition

An (n, k, c)-expander: Xn,k = (V , E) with expansion coefficient c = inf |∂(F)| |F| : F ⊆ V , 0 < |F| ≤ n 2

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

• λ(X) = max{|λ(X)| : λ ∈ Spec(X), |λ| = k}
• ∂(F) = {{u, v} ∈ E | u ∈ F, v ∈ V \ F} (boundary of F)

Definition

An (n, k, c)-expander: Xn,k = (V , E) with expansion coefficient c = inf |∂(F)| |F| : F ⊆ V , 0 < |F| ≤ n 2

Expander Graphs

• If (Xn,k) is a k-regular graph on n vertices, the eigenvalues of

A are k ≥ λ1 ≥ λ2 ≥ . . . ≥ λn−1 ≥ −k

• λ(X) = max{|λ(X)| : λ ∈ Spec(X), |λ| = k}
• ∂(F) = {{u, v} ∈ E | u ∈ F, v ∈ V \ F} (boundary of F)

Definition

An (n, k, c)-expander: Xn,k = (V , E) with expansion coefficient c = inf |∂(F)| |F| : F ⊆ V , 0 < |F| ≤ n 2

• Want large expansion coefficient.

Good expanders/Ramanujan Graphs

Proposition (Cheeger’s inequality)

Connected Xn,k is an (n, k, c)-expander with c ≥ k − λ(Xn,k) 2 .

Good expanders/Ramanujan Graphs

Proposition (Cheeger’s inequality)

Connected Xn,k is an (n, k, c)-expander with c ≥ k − λ(Xn,k) 2 .

Theorem (Alon-Boppana)

lim inf

n→∞ λ(Xn,k) ≥ 2

√ k − 1

Good expanders/Ramanujan Graphs

Proposition (Cheeger’s inequality)

Connected Xn,k is an (n, k, c)-expander with c ≥ k − λ(Xn,k) 2 .

Theorem (Alon-Boppana)

lim inf

n→∞ λ(Xn,k) ≥ 2

√ k − 1

Definition (Lubotzky, Phillips, Sarnak)

A k-regular graph X is a Ramanujan graph if |λ(X)| ≤ 2 √ k − 1.

Good expanders/Ramanujan Graphs

Proposition (Cheeger’s inequality)

Connected Xn,k is an (n, k, c)-expander with c ≥ k − λ(Xn,k) 2 .

Theorem (Alon-Boppana)

lim inf

n→∞ λ(Xn,k) ≥ 2

√ k − 1

Definition (Lubotzky, Phillips, Sarnak)

A k-regular graph X is a Ramanujan graph if |λ(X)| ≤ 2 √ k − 1.

Corollary (to Ihara’s Theorem)

X satisfies the Riemann hypothesis ⇐ ⇒ X is a Ramanujan graph

Constructing Ramanujan Graphs

• infinite family of Ramanujan graphs Xn,k, k fixed, n → ∞

Constructing Ramanujan Graphs

• infinite family of Ramanujan graphs Xn,k, k fixed, n → ∞
• for large n hard to estimate eigenvalues of A

Constructing Ramanujan Graphs

• infinite family of Ramanujan graphs Xn,k, k fixed, n → ∞
• for large n hard to estimate eigenvalues of A
• attach a graph to a group

Constructing Ramanujan Graphs

• infinite family of Ramanujan graphs Xn,k, k fixed, n → ∞
• for large n hard to estimate eigenvalues of A
• attach a graph to a group
• group structure makes estimation of eigenvalues possible

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx
• H generated by T = χ(K) and Tp = χ
• K

1 p

• K

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx
• H generated by T = χ(K) and Tp = χ
• K

1 p

• K
• H acts on L2(G/K) by (f • φ)(x) :=
• G φ(y)f (xy) dy

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx
• H generated by T = χ(K) and Tp = χ
• K

1 p

• K
• H acts on L2(G/K) by (f • φ)(x) :=
• G φ(y)f (xy) dy
• Tp acts on L2(V ) = L2(G/K) as adjacency operator

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx
• H generated by T = χ(K) and Tp = χ
• K

1 p

• K
• H acts on L2(G/K) by (f • φ)(x) :=
• G φ(y)f (xy) dy
• Tp acts on L2(V ) = L2(G/K) as adjacency operator

Graphs from Groups

• graph X = (V , E): Bruhat-Tits building of G = GL2(Qp)
• one vertex for each conjugate of K = GL2(Zp); V ↔ G/K
• (p + 1)-regular tree × affine line
• G acts simplicially on X
• Hecke algebra: H(G, K) = {f : G → C |f (kgk′) = f (g)}C

(φ ∗ ψ)(g) :=

• G φ(x)ψ(x−1g) dx
• H generated by T = χ(K) and Tp = χ
• K

1 p

• K
• H acts on L2(G/K) by (f • φ)(x) :=
• G φ(y)f (xy) dy
• Tp acts on L2(V ) = L2(G/K) as adjacency operator

(Generalize to GLn(Qp): B. 2001.)

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H
• ne dimensional representations of H

1–1 (Satake isomorphism)

irreducible unramified (spherical) representations of G

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H
• ne dimensional representations of H

1–1 (Satake isomorphism)

irreducible unramified (spherical) representations of G

• Γ\X Ramanujan ⇔ all irreducible unramified (unitary)

representations of G appearing in L2(G/Γ) are tempered (RC).

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H
• ne dimensional representations of H

1–1 (Satake isomorphism)

irreducible unramified (spherical) representations of G

• Γ\X Ramanujan ⇔ all irreducible unramified (unitary)

representations of G appearing in L2(G/Γ) are tempered (RC).

Theorem

Graph obtained above is Ramanujan

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H
• ne dimensional representations of H

1–1 (Satake isomorphism)

irreducible unramified (spherical) representations of G

• Γ\X Ramanujan ⇔ all irreducible unramified (unitary)

representations of G appearing in L2(G/Γ) are tempered (RC).

Theorem

Graph obtained above is Ramanujan

Eigenvalues of A

• λ = λi ∈ Spec(X) ⇐

⇒ Ti(f ) = λi · f

• λ (one dimensional) representation of H
• ne dimensional representations of H

1–1 (Satake isomorphism)

irreducible unramified (spherical) representations of G

• Γ\X Ramanujan ⇔ all irreducible unramified (unitary)

representations of G appearing in L2(G/Γ) are tempered (RC).

Theorem

Graph obtained above is Ramanujan

Ramanujan Graphs and Generalizations

• n = 2: LPS (Jacquet-Langlands correspondence)

Ramanujan Graphs and Generalizations

• n = 2: LPS (Jacquet-Langlands correspondence)
• n = 3: B. (Rogawski’s classification of representations of U(3)

& Arthur’s conjectures)

Ramanujan Graphs and Generalizations

• n = 2: LPS (Jacquet-Langlands correspondence)
• n = 3: B. (Rogawski’s classification of representations of U(3)

& Arthur’s conjectures)

• general n (function filed case): Lubotzky-Samuels-Vishne

(Laforgue’s work)

Ramanujan Graphs and Generalizations

• n = 2: LPS (Jacquet-Langlands correspondence)
• n = 3: B. (Rogawski’s classification of representations of U(3)

& Arthur’s conjectures)

• general n (function filed case): Lubotzky-Samuels-Vishne

(Laforgue’s work)

• Ramanujan hypergraphs: Li (Laumot-Rapoport-Stuhler)

Ramanujan Bigraphs (w. Dan Ciubotaru)

Bk,l,n: (k, l)-regular bigraph (biregular, bipartite) on n vertices. Largest eigenvalue: √ kl.

Ramanujan Bigraphs (w. Dan Ciubotaru)

Bk,l,n: (k, l)-regular bigraph (biregular, bipartite) on n vertices. Largest eigenvalue: √ kl.

Theorem (Li, Feng)

lim inf

n→∞ |λ(Bk,l,n)| ≥

√ k − 1 + √ l − 1

Ramanujan Bigraphs (w. Dan Ciubotaru)

Bk,l,n: (k, l)-regular bigraph (biregular, bipartite) on n vertices. Largest eigenvalue: √ kl.

Theorem (Li, Feng)

lim inf

n→∞ |λ(Bk,l,n)| ≥

√ k − 1 + √ l − 1

Definition (Hashimoto)

A (q1 + 1, q2 + 1)-bigraph is called Ramanujan bigraph if |λ2 − q1 − q2| ≤ 2√q1q2, ∀ λ ∈ Spec(X), λ2 = (1 + q1)(1 + q2).

Ramanujan Bigraphs (w. Dan Ciubotaru)

Bk,l,n: (k, l)-regular bigraph (biregular, bipartite) on n vertices. Largest eigenvalue: √ kl.

Theorem (Li, Feng)

lim inf

n→∞ |λ(Bk,l,n)| ≥

√ k − 1 + √ l − 1

Definition (Hashimoto)

A (q1 + 1, q2 + 1)-bigraph is called Ramanujan bigraph if |λ2 − q1 − q2| ≤ 2√q1q2, ∀ λ ∈ Spec(X), λ2 = (1 + q1)(1 + q2).

Definition

A (q1 + 1, q2 + 1)-bigraph satisfies RH if Re(s) ∈ (0, 1) and ZX((q1q2)−s))−1 = 0 ⇒ Re(s) = 1 2

Weak Ramanujan Bigraphs

• X finite, connected, (q1 + 1, q2 + 1)-bigraph with q1 ≥ q2

Weak Ramanujan Bigraphs

• X finite, connected, (q1 + 1, q2 + 1)-bigraph with q1 ≥ q2
• ni number of vertices with valency qi + 1 (then, n2 ≥ n1)

Weak Ramanujan Bigraphs

• X finite, connected, (q1 + 1, q2 + 1)-bigraph with q1 ≥ q2
• ni number of vertices with valency qi + 1 (then, n2 ≥ n1)
• Spec(X) = {±λ1, ±λ2, . . . ± λn1, 0, . . . , 0

n2−n1

} λ1 =

• (1 + q1)(1 + q2) > λ2 ≥ · · · ≥ λn1 ≥ 0

Weak Ramanujan Bigraphs

• X finite, connected, (q1 + 1, q2 + 1)-bigraph with q1 ≥ q2
• ni number of vertices with valency qi + 1 (then, n2 ≥ n1)
• Spec(X) = {±λ1, ±λ2, . . . ± λn1, 0, . . . , 0

n2−n1

} λ1 =

• (1 + q1)(1 + q2) > λ2 ≥ · · · ≥ λn1 ≥ 0

Definition

A (q1 + 1, q2 + 1)-bigraph X is a weak Ramanujan bigraph if λn1 > 0, i.e., 0 has multiplicity exactly n2 − n1 in Spec(X).

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)
• ˜

X - Bruhat-Tits building of G (q1 + 1, q2 + 1) = (p3 + 1, p + 1) biregular, bipartite tree

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)
• ˜

X - Bruhat-Tits building of G (q1 + 1, q2 + 1) = (p3 + 1, p + 1) biregular, bipartite tree

• X = ˜

X/Γ, Γ discrete, co-compact subgroup of G

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)
• ˜

X - Bruhat-Tits building of G (q1 + 1, q2 + 1) = (p3 + 1, p + 1) biregular, bipartite tree

• X = ˜

X/Γ, Γ discrete, co-compact subgroup of G

• Maximal compact subgroups of G: {K1}, {K2}.

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)
• ˜

X - Bruhat-Tits building of G (q1 + 1, q2 + 1) = (p3 + 1, p + 1) biregular, bipartite tree

• X = ˜

X/Γ, Γ discrete, co-compact subgroup of G

• Maximal compact subgroups of G: {K1}, {K2}.
• I = K1 ∩ K2: Iwahori subgroup (fixes an edge)

Bigraphs from groups

• G = SU(3) (def. by unramified sep. quadratic ext. of Qp)
• ˜

X - Bruhat-Tits building of G (q1 + 1, q2 + 1) = (p3 + 1, p + 1) biregular, bipartite tree

• X = ˜

X/Γ, Γ discrete, co-compact subgroup of G

• Maximal compact subgroups of G: {K1}, {K2}.
• I = K1 ∩ K2: Iwahori subgroup (fixes an edge)
• H(G, I) = T1, T2/(T 2

i = (qi − 1)Ti + qi)

Adjacency and the Iwahori-Hecke algebra

• T1, T2 endomorphisms on C[E(X)],

(Tif )(e) :=

• e′∈Ei(e)

f (e′) − f (e) (i = 1, 2).

Adjacency and the Iwahori-Hecke algebra

• T1, T2 endomorphisms on C[E(X)],

(Tif )(e) :=

• e′∈Ei(e)

f (e′) − f (e) (i = 1, 2).

• (f , f ′) =
• e∈E(X)

(f (e), f ′(e))C

Adjacency and the Iwahori-Hecke algebra

• T1, T2 endomorphisms on C[E(X)],

(Tif )(e) :=

• e′∈Ei(e)

f (e′) − f (e) (i = 1, 2).

• (f , f ′) =
• e∈E(X)

(f (e), f ′(e))C

Theorem (Hashimoto)

ZX(u)−1 = det(I − (T1T2)u) = (1 − u)r−1(1 + q2u)n2−n1

n1

• j=1

(1 − (λ2

j − q1 − q2)u + q1q2u2)

Adjacency and the Iwahori-Hecke algebra

• T1, T2 endomorphisms on C[E(X)],

(Tif )(e) :=

• e′∈Ei(e)

f (e′) − f (e) (i = 1, 2).

• (f , f ′) =
• e∈E(X)

(f (e), f ′(e))C

Theorem (Hashimoto)

ZX(u)−1 = det(I − (T1T2)u) = (1 − u)r−1(1 + q2u)n2−n1

n1

• j=1

(1 − (λ2

j − q1 − q2)u + q1q2u2)

• Iwahori-Hecke algebra H = H(G, I) ∼

= C[T1, T2]

Representations of the Iwahori-Hecke Algebra (Hashimoto)

• Finite dim’l irreducible representations ϕ of H have dimension 1
• r 2 and they are determined by the characteristic polynomial pϕ
• f ϕ(T1T2).
• Degree two irreducible representations are parameterized by

c ∈ C, c = 0, c = (q1 + 1)(q2 + 1). pϕ(u) = det(1 − ϕ(T1T2)u) = 1 − (c − q1 − q2)u + q1q2u2.

• One dimensional irreducible representation have characteristic

polynomial pSt(u) = 1 − u; pds(u) = 1 + q2u; psph(u) = 1 − q1q2u; pnt(u) = 1 + q1u.

Representations (continued)

c ∈ Spec(X)

• ϕ ↔ π sherical unitary irreducible representation of G appearing in

L2(G/Γ)

• mult. of pϕ(u) in ZX(u)−1 = mult. of π in L2(G/Γ)I =: m(π)

Iwahori-Hecke Algebra

• (Bernstein-Lusztig presentation) H = HW ⊗ A

HW = C[T]/T 2 = (z2λ − 1)T + z2λ, A = C[θ] commutation relation: θT − Tθ−1 = (z2λ − 1)θ + (zλ+λ∗ − zλ−λ∗) (for SU3(Qp), z = √p, λ = 3, λ∗ = 1)

Iwahori-Hecke Algebra

• (Bernstein-Lusztig presentation) H = HW ⊗ A

HW = C[T]/T 2 = (z2λ − 1)T + z2λ, A = C[θ] commutation relation: θT − Tθ−1 = (z2λ − 1)θ + (zλ+λ∗ − zλ−λ∗) (for SU3(Qp), z = √p, λ = 3, λ∗ = 1)

• Iwahori ↔ Berenstein-Lusztig

T = T1, θ =

1 √q1q2 T1T2,

z2λ = q1 and z2λ∗ = q2 pϕ(u) = det(1−√q1q2ϕ(θ)u) = 1−√q1q2Tr(ϕ(θ))u+q1q2u2

Tempered representations

If m(π) > 0 and ϕ ∈ {St, ds}, Tr(ϕ(Θ)) = 1 √q1q2 Tr(ϕ(T1T2)) = c − q1 − q2 √q1q2

Tempered representations

If m(π) > 0 and ϕ ∈ {St, ds}, Tr(ϕ(Θ)) = 1 √q1q2 Tr(ϕ(T1T2)) = c − q1 − q2 √q1q2 Ramanujan condition |c − q1 − q2| ≤ 2√q1q2

• |Tr(ϕ(Θ))| ≤ 2 (i.e., ϕ is tempered)

Ramanujan bigraphs

Ramanujan Type Conjecture: Every nontrivial irreducible unitary H(G, I)-module that appears in the decomposition of L2(G/Γ)I = L2(I\G/Γ) is tempered.

Ramanujan bigraphs

Ramanujan Type Conjecture: Every nontrivial irreducible unitary H(G, I)-module that appears in the decomposition of L2(G/Γ)I = L2(I\G/Γ) is tempered.

Theorem (B.-C.)

G = SU(3). Γ ≤ G discrete, co-compact, acts on G without fixed points. ˜ X is the Bruhat-Tits tree of G Then, X = ˜ X/Γ is a Ramanujan bigraph if and only if G satisfies the Ramanujan Type Conjecture.

Ramanujan bigraphs

Ramanujan Type Conjecture: Every nontrivial irreducible unitary H(G, I)-module that appears in the decomposition of L2(G/Γ)I = L2(I\G/Γ) is tempered.

Theorem (B.-C.)

G = SU(3). Γ ≤ G discrete, co-compact, acts on G without fixed points. ˜ X is the Bruhat-Tits tree of G Then, X = ˜ X/Γ is a Ramanujan bigraph if and only if G satisfies the Ramanujan Type Conjecture.

Ramanujan bigraphs

Ramanujan Type Conjecture: Every nontrivial irreducible unitary H(G, I)-module that appears in the decomposition of L2(G/Γ)I = L2(I\G/Γ) is tempered.

Theorem (B.-C.)

G = SU(3). Γ ≤ G discrete, co-compact, acts on G without fixed points. ˜ X is the Bruhat-Tits tree of G Then, X = ˜ X/Γ is a Ramanujan bigraph if and only if G satisfies the Ramanujan Type Conjecture.

Corollary

The graph X is Ramanujan if and only if X is weakly Ramanujan.

Constructions

Theorem (Rogawski)

If G is a compact inner form of U(3) arising from a division algebra with an involution of the second kind, there are no non-tempered representations (the Ramanujan-Petersson conjecture is satisfied).