The building X v on which G ( k v ) and G ( k v ) act, when v splits - PowerPoint PPT Presentation

The building X v on which G ( k v ) and ¯ G ( k v ) act, when v splits in ℓ . K := nonarchimedean local field, with valuation v . o K := { x ∈ K : v ( x ) ≥ 0 } . { x ∈ K : v ( x ) > 0 } equals π o K . q := | o K /π o K | . When K = k v , write o v for o k v , q v for q . 1

Any basis { v 1 , v 2 , v 3 } of K gives a lattice in K 3 : L = { a 1 v 1 + a 2 v 2 + a 3 v 3 : a 1 , a 2 , a 3 ∈ o K } . (1) E.g. { v 1 , v 2 , v 3 } = { e 1 , e 2 , e 3 } gives L 0 := o 3 K . Lat K := set of lattices in K 3 . g ∈ GL (3 , K ) & L ∈ Lat K ⇒ g ( L ) ∈ Lat K . GL (3 , K ) acts transitively on Lat K . GL (3 , o K ) := { g ∈ GL (3 , K ) : g & g − 1 have entries in o K } . GL (3 , o K ) equals { g ∈ GL (3 , K ) : g ( L 0 ) = L 0 } . GL (3 , o K ) = { g ∈ M 3 × 3 ( o K ) : v (det( g )) = 0 } . 2

L 1 , L 2 ∈ Lat K equivalent if L 2 = t L 1 , some t ∈ K × . [ L ] := equivalence class of L . X K := set of equivalence classes. For g ∈ GL (3 , K ), g. [ L ] := [ g ( L )]. GL (3 , K ) acts transitively on X K . g = tI ⇒ g. [ L ] = [ L ] for all L ∈ Lat K . PGL (3 , K ) acts transitively on X K . For i ∈ { 0 , 1 , 2 } , [ g ( L 0 )] ∈ X K has type i if v (det( g )) ≡ i (mod 3). SL (3 , K ) acts transitively on { [ L ] ∈ X K : type([ L ]) = i } . 3

[ L 1 ] is adjacent to [ L 2 ] if there are representatives L j of [ L j ] for j = 1 , 2 so that π L 1 � L 2 � L 1 . This implies π L 2 � π L 1 � L 2 , so adjacency is a symmetric relation. Adjacent lattice classes have dif- ferent types. Fact: Given [ L ] with type([ L ]) = i , and j � = i , ♯ { [ M ] ∈ X K : [ M ] adjacent to [ L ] & type([ M ]) = j } = q 2 + q + 1 . Proof: L /π L is a vector space of dimension 3 over the residual field o K /π o K . For ν = 1 , 2, π L ⊂ M ⊂ L and type([ M ]) = i + ν (mod 3) iff M /π L is a ν -dimensional subspace. 4

[ L 1 ] , [ L 2 ] , [ L 3 ] form a chamber if there are representatives L j of [ L j ] for j = 1 , 2 , 3 so that π L 1 � L 3 � L 2 � L 1 . Each chamber contains one lattice class of each type. Any pair of adjacent lattice classes lies in q + 1 distinct chambers. Any lattice class belongs to ( q 2 + q + 1)( q + 1) distinct chambers. X K is a simplicial complex. 5

For g ∈ SL (3 , K ) and L ∈ Lat K , g. [ L ] = [ L ] iff g ( L ) = L . For L ∈ Lat K , { g ∈ SL (3 , K ) : g ( L ) = L} is a maximal compact subgroup of SL (3 , K ). Any maximal compact subgroup of SL (3 , K ) has this form. There are three conjugacy classes of maximal compact subgroups of SL (3 , K ), corresponding to the three types. Any two maximal compact subgroups of SL (3 , K ) are conjugate by an element of GL (3 , K ). 6

For i = 1 , 2, SL (3 , o K ) acts transitively on { [ L ] ∈ X K : [ L ] adjacent to [ L 0 ] & type([ L ]) = i } . For any edge containing [ L 0 ], the stabilizer in SL (3 , K ) of that edge has index q 2 + q + 1 in SL (3 , o K ). SL (3 , o K ) acts transitively on the set of chambers containing [ L 0 ]. For any chamber containing [ L 0 ], the stabilizer in SL (3 , K ) of that chamber has index ( q 2 + q + 1)( q + 1) in SL (3 , o K ). 7

We have seen that when v ∈ V f \ T 0 splits in ℓ , then G ( k v ) ∼ = SL (3 , k v ). The parahoric subgroups of G ( k v ) are its subgroups corresponding to the stabilizers of vertices, edges and chambers of X v := X k v . In particular, P v = { g ∈ SL (3 , k v ) : g ( o 3 v ) = o 3 v } = SL (3 , o v ) . is a maximal parahoric subgroup of G ( k v ). Given ξ ∈ D satisfying ι ( ξ ) ξ = 1, we need to understand when the image ξ v of ξ in GL (3 , k v ) under the map D → D ⊗ ℓ k v ∼ = M 3 × 3 ( k v ) fixes the lattice class [ o 3 v ]. In particular, if ξ ∈ G ( k ) when does ξ v fix o 3 v ? 8

b 1 , . . . , b 6 := basis of m over k . a ij b i σ j . � ξ ∈ D ⇒ ξ = i,j Form matrix B = ( ψ i ( b j )), where Gal( m/k ) = { ψ 1 , . . . , ψ 6 } . Fact: det( B ) 2 ∈ ℓ . Lemma. b 1 , . . . , b 6 can be chosen so that v (det( B ) 2 ) = 0 for all v ∈ V f \T 0 which split in ℓ . Example. In ( a = 7 , p = 2) case, m = Q [ ζ ], where ζ = ζ 7 . Choose b 1 , . . . , b 6 = 1 , ζ, . . . , ζ 5 . Then det( B ) = 7 2 s for s = 1 + 2 ζ + 2 ζ 2 + 2 ζ 4 , so det( B ) 2 = − 7 5 . So v (det( B ) 2 ) = 0 unless v = u 7 . 9

Proposition. If ι ( ξ ) ξ = 1, and v ∈ V f \ T 0 splits in ℓ , then ξ v ( o 3 v ) = o 3 ⇔ a ij ∈ k ∩ o v for all i, j. v When m ֒ → k v , the calculations only involve the matrix B . When m � ֒ → k v , we also need the following: Lemma. If v ∈ V f \ T 0 splits in ℓ , but m � ֒ → k v , there is an η v ∈ k v ( Z ) such that N k v ( Z ) /k v ( η v ) = D . Moreover, ˜ v ( η v ) = 0. → k v ( Z ). This gives Reasons: Embed ℓ in k v , then extend to m = ℓ ( Z ) ֒ valuation ˜ v on k v ( Z ) extending v . The image of D ∈ ℓ in k v satisfies v ( D ) = 0. Case by case we see k v ( Z ) is an unramified extension of k v . Now apply norm theorem from local class field theory. 10

→ k v , the isomorphism D ⊗ ℓ k v ∼ When v ∈ V f \ T 0 splits in ℓ but m � ֒ = M 3 × 3 ( k v ) involves conjugation by J v = Θ C v . Here C v is diagonal matrix with diagonal entries η v , 1 and 1 /ϕ ( η v ), and Θ = ( ϕ i ( θ j )) for some basis θ 0 , θ 1 , θ 2 of m over ℓ . Lemma. θ 0 , θ 1 , θ 2 can be chosen in o m and so that ˜ v (det(Θ)) = 0 for all v ∈ V f \ T 0 which split in ℓ . In ( a = 7 , p = 2) case, m = Q [ ζ ], where ζ = ζ 7 . Example. Choose θ 0 , θ 1 , θ 2 = 1 , ζ, ζ 2 . Then det(Θ) = − s . So ˜ v (Θ) = 0 unless v = u 7 . 11

The building X v when v does not split in ℓ . K := non-archimedean local field, with valuation v , as before. L := K ( s ), a separable quadratic extension of K . The automorphism a + bs �→ a − bs of L is denoted x �→ ¯ x . ˜ v := unique extension to L of v . o L := { x ∈ L : ˜ v ( x ) ≥ 0 } , and { x ∈ L : ˜ v ( x ) > 0 } equals π L o L . When K = k v , where v ∈ V f does not split in ℓ , L = k v ( s ) is the com- pletion ℓ ˜ v of ℓ with respect to the unique extension ˜ v of v to ℓ , and we write o ˜ v for o L . 12

f : L 3 × L 3 → L : a nondegenerate sesquilinear form on L 3 . Then f ↔ a nonsingular Hermitian matrix F : f ( x, y ) = y ∗ F x , where x = x 1 e 1 + x 2 e 2 + x 3 e 3 and y = y 1 e 1 + y 2 e 2 + y 3 e 3 . The unitary group of F is U F = { g : L 3 → L 3 : f ( gx, gy ) = f ( x, y ) for all x, y ∈ L 3 } . U F ∼ = { g ∈ M 3 × 3 ( L ) : g ∗ Fg = F } . SU F ∼ = { g ∈ M 3 × 3 ( L ) : g ∗ Fg = F and det( g ) = 1 } . 13

For L ∈ Lat L , L ′ = { x ∈ L 3 : f ( x, y ) ∈ o L for all y ∈ L} (2) is again a lattice, called the dual lattice of L with respect to f . Then ( L ′ ) ′ = L L ′ 2 ⊂ L ′ and L 1 ⊂ L 2 ⇐ ⇒ 1 . For L 0 = o 3 L and g ∈ GL (3 , L ), ( g ( L 0 )) ′ = ( g ∗ F ) − 1 ( L 0 ) . (3) 14

Let Lat 1 = {L ∈ Lat L : L ′ = L} , and Lat 2 = { ( M , M ′ ) : M ∈ Lat L and π M ′ � M � M ′ } . ( M , M ′ ) ∈ Lat 2 iff the lattice classes [ M ] and [ M ′ ] are adjacent in the building X L of SL (3 , L ). L ∈ Lat 1 and ( M , M ′ ) ∈ Lat 2 are called adjacent if π L � M � L . This means that π L � π M ′ � M � L , so that [ L ], [ M ] and [ M ′ ] form a chamber in X L . 15

Using (3), we see that L 0 ∈ Lat 1 ⇔ F ∈ GL (3 , o L ) . If g ∈ U F and L ∈ Lat L , then • ( g ( L )) ′ = g ( L ′ ), • g fixes L iff g fixes L ′ , • if L ∈ Lat 1 , then g ( L ) ∈ Lat 1 , and • if ( M , M ′ ) ∈ Lat 2 , then ( g ( M ) , g ( M ′ )) ∈ Lat 2 . So U F acts on each of the sets Lat 1 and Lat 2 . 16

Suppose that F ∈ GL (3 , o L ) so that L 0 = o 3 Lemma. L is in Lat 1 . Let g ∈ GL (3 , K ), and M = g ( L 0 ). Then ( M , M ′ ) is in Lat 2 , and is adjacent to L 0 if and only if (a) If g has entries in o L , (b) If πg − 1 has entries in o L , (c) If g ∗ Fg has entries in o L , (d) If π ( g ∗ Fg ) − 1 has entries in o L , (e) If ˜ v (det( g )) = ˜ v ( π ). 17

Lemma. Let g ∈ U F . Then (a) If L ∈ Lat 1 , then g ( L ) = L if and only if g ( L ) ⊂ L . (b) If ( M , M ′ ) ∈ Lat 2 , then g ( M ) = M if and only if g ( M ) ⊂ M . (c) If ( M , M ′ ) ∈ Lat 2 , then g ( M ′ ) = M ′ if and only if g ( M ′ ) ⊂ M ′ . 18

There is a building B associated with SU F . This is a very special case of results of Bruhat and Tits (see § 10.1 in Bruhat-Tits “Groupes R´ eductifs sur un corps local I: Donn´ ees radicielles valu´ ees”, Publ. Math. I.H.E.S. 41 (1972), 5–251, and § 1.15 in Tits, “Reductive groups over local fields”, Proc. 33 (1979), Amer. Math. Soc. Symp. Pure Math. 29–69. In this case, the building B is a tree. Theorem. With the above notation, the set Lat 1 ∪ Lat 2 , together with the above adjacency relation, forms a tree T . This tree is homogeneous of degree q +1 when L is a ramified extension of K , and is bihomogeneous when L is an unramified extension of K , each v ∈ Lat 1 having q 3 + 1 neighbors, and each v ∈ Lat 2 having q + 1 neighbors. It is isomorphic to the Bruhat Tits building B associated with SU F . Elements of Lat i are called vertices of type i of this tree. 19