Symbolic calculus on coadjoint orbits of nilpotent Lie groups - PowerPoint PPT Presentation

Symbolic calculus on coadjoint orbits of nilpotent Lie groups Daniel Beltit ¸˘ a Institute of Mathematics of the Romanian Academy *** Joint work with Ingrid Beltit ¸˘ a (IMAR) Bia� lowie˙ za, June 27, 2012 Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 1 / 16

References ◮ I. Beltit ¸˘ a, D. B. , Boundedness for Weyl-Pedersen calculus on flat coadjoint orbits. Preprint arXiv:1203.0974v1 [math.AP]. ◮ I. Beltit ¸˘ a, D. B. , Modulation spaces of symbols for representations of nilpotent Lie groups. J. Fourier Analysis Appl. 17 (2011), no. 2, 290–319. ◮ I. Beltit ¸˘ a, D. B. , Algebras of symbols associated with the Weyl calculus for Lie group representations. Monatshefte f¨ ur Math. (to appear). ◮ I. Beltit ¸˘ a, D. B. , Continuity of magnetic Weyl calculus. J. Functional Analysis 260 (2011), no. 7, 1944-1968. Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 2 / 16

Plan 1 Early origins of the pseudo-differential Weyl calculus: H. Weyl (1928), M.E. Taylor (1968), R.F.V. Anderson (1969), L. H¨ ormander (1979) 2 Orbital Weyl calculus for unirreps of nilpotent Lie groups: N.V. Pedersen (1994) 3 Continuity of pseudo-differential operators Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 3 / 16

Origins of the pseudo-differential Weyl calculus Quantum positions Q 1 , . . . , Q n and momenta P 1 , . . . , P n ( Q j f )( q ) = q j f ( q ) and P j f = 1 ∂ f for j = 1 , . . . , n . ∂ q j i Fact: for p = ( p 1 , . . . , p n ) ∈ R n and q = ( q 1 , . . . , q n ) ∈ R n , q · Q + p · P := q 1 Q 1 + · · · + q n Q n + p 1 P 1 + · · · + p n P n is a self-adjoint oper. in L 2 ( R n ), so there exists the unitary e i ( q · Q + p · P ) . ormander calculus a ❀ a ( Q , P ) (symbol ❀ pseudo-diff. oper.) Weyl-H¨ �� e i ( q · Q + p · P ) : S ( R n ) → S ′ ( R n ) a ( Q , P ) = a ( q , p ) d p d q � � �� � π ( q , p , 0) R n × R n “Integral” kernel formula: �� � q + q ′ � e i ( q − q ′ ) · p f ( q ′ ) d p d q ′ ( a ( Q , P ) f )( q ) = a , p 2 R n × R n Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 4 / 16

Schr¨ odinger repres. of the Heisenberg group Canonical commutation relations [ Q j , Q k ] = [ P j , P k ] = 0 and [ Q j , P k ] = δ jk i · I for j , k = 1 , . . . , n • Heisenberg algebra : h 2 n +1 = R n × R n × R p · q ′ − p ′ · q [( q , p , t ) , ( q ′ , p ′ , t ′ )] = [(0 , 0 , )] � �� � symplectic form on R n × R n • Heisenberg group : H 2 n +1 = ( h 2 n +1 , ∗ ), X ∗ Y := log( e X e Y ) = X + Y + 1 2[ X , Y ]; unit element 0 ∈ H 2 n +1 , inversion X − 1 := − X . odinger representation π : H 2 n +1 → B ( L 2 ( R n )), • Schr¨ π ( q , p , t ) f ( x ) = e i ( px + 1 2 pq + t ) f ( q + x ) Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 5 / 16

Orbital Weyl calculus for nilpotent Lie groups, pag. 1 (classical phase space) • G nilpotent Lie group with Lie alg. g and pairing �· , ·� : g ∗ × g → R hence [ g , [ g , . . . [ g , g ] . . . ]] = { 0 } ; e.g., [ h 2 n +1 , [ h 2 n +1 , h 2 n +1 ]] = { 0 } . → g ∗ • coadjoint orbit O ֒ • ξ 0 ∈ O ❀ g ξ 0 = { X ∈ g | ξ 0 ◦ ad g X = 0 } • { 0 } = g 0 ⊂ g 1 ⊂ · · · ⊂ g m = g Jordan-H¨ older sequence ❀ the set of jump indices e ⊂ { 1 , . . . , n } for O • X j ∈ g j \ g j − 1 for j = 1 , . . . , m ❀ g e = span { X j | j ∈ e } ⊆ g ❀ O ∼ → g ∗ e , ξ �→ ξ | g e � O e − i � ξ, •� a ( ξ ) d ξ • Fourier tr. a �→ ˆ a = S ( O ) ∼ → S ( g e ), L 2 ( O ) ∼ → L 2 ( g e ) Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 6 / 16

Orbital Weyl calculus for nilpotent Lie groups, pag. 2 (quantization of the phase space) • π : G → B ( H ) unirrep assoc. with O • H ∞ = { φ ∈ H | π ( · ) φ ∈ C ∞ ( G , H ) } space of smooth vectors H −∞ (=anti-dual of H ∞ ) sp. of distribution vectors (e.g., S ( R n ) ֒ → L 2 ( R n ) ֒ → S ′ ( R n )) ❀ H ∞ ֒ → H ֒ → H −∞ • π ⊗ 2 : G × G → B ( S 2 ( H )), π ⊗ 2 ( g 1 , g 2 ) T = π ( g 1 ) T π ( g 2 ) − 1 ❀ B ( H ) ∞ = { T ∈ S 2 ( H ) | π ⊗ 2 ( · ) T ∈ C ∞ ( G × G , S 2 ( H )) } ֒ → S 1 ( H ) � �� � ≃L ( H −∞ , H ∞ ) regularizing opers. Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 7 / 16

Orbital Weyl calculus for nilpotent Lie groups, pag. 3 (quantization of the observables) g ∗ ⊃ O ← → π : G → B ( H ) Weyl-Pedersen calculus (for a ∈ S ′ ( O )) � Op π ( a ) = g e π (exp G X )ˆ a ( X ) d X : H ∞ → H −∞ Properties (N.V. Pedersen, 1994) ◮ Op π : S ( O ) ∼ → B ( H ) ∞ , S ′ ( O ) ∼ → L ( H ∞ , H −∞ ), L 2 ( O ) ∼ → S 2 ( H ) ◮ Op π ( �· , X �| O ) = d π ( X ) for X ∈ g � ◮ Tr(Op π ( a )) = O a ( ξ ) d ξ ◮ Op π (¯ a ) = Op π ( a ) ∗ Earlier results: A. Melin (1981–1983), K.G. Miller (1982), D. Manchon (1991) Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 8 / 16

Analysis of the square-integrable repres., pag. 1 • Z the center of the group G • repres. π square integrable mod. Z with the coadjoint orbit O � G / Z | ( π ( · ) ψ | ψ ) | 2 < ∞ Definition: ( ∃ ψ ∈ H \ { 0 } ) Characterization: O is an affine subspace in g ∗ odinger repres. is R 2 n × { 1 } ֒ → R 2 n +1 . Example: The orbit of the Schr¨ Theorem (covariance of the Weyl-Pedersen calculus) a ∈ S ′ ( O ), g ∈ G ⇒ Op ( a ◦ Ad ∗ G ( g ) | O ) = π ( g ) − 1 Op ( a ) π ( g ) Diff ( O ) = the set of all linear differential operators D : C ∞ ( O ) → C ∞ ( O ) which are invariant to the coadjoint action: ( ∀ g ∈ G )( ∀ a ∈ C ∞ ( O )) D ( a ◦ Ad ∗ G ( g ) | O ) = ( Da ) ◦ Ad ∗ G ( g ) | O . Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 9 / 16

Analysis of the square-integrable repres., pag. 2   all compact operators if p = 0 ,  { T ∈ B ( H ) | trace | T | p < ∞} S p ( H ) = if 1 ≤ p < ∞ ,   B ( H ) if p = ∞ . Main Theorem Assume dim O = dim G − dim Z and p ∈ { 0 } ∪ [1 , ∞ ]. 1 Equivalent assertions for a ∈ C ∞ ( O ): ◮ for every D ∈ Diff ( O ) we have Da ∈ L p ( O ), not.: a ∈ C ∞ , p ( O ); ◮ for every D ∈ Diff ( O ) we have Op ( Da ) ∈ S p ( H ). 2 C ∞ , p ( O ) has a Fr´ echet topology defined by any of the families of seminorms { a �→ � Da � L p ( O ) } D ∈ Diff ( O ) and { a �→ � Op ( Da ) � S p ( H ) } D ∈ Diff ( O ) . 3 Weyl-Pedersen calculus Op : C ∞ , p ( O ) → S p ( H ) is cont. linear. Here L 0 ( O ) := { a ∈ L ∞ ( O ) | lim ξ →∞ a ( ξ ) = 0 } . Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 10 / 16

Ingredients of the proof • g 0 := g / z ❀ G 0 acts simply transitively on O • Convolution w.r.t. the coadjoint action : b ∈ L 1 loc ( G 0 ), a ∈ L ∞ ( O ) with compact support, � b ( T ) a ( Ad ∗ ( − T ) ξ ) d T ( b ∗ a )( ξ ) = for ξ ∈ O . G 0 This gives � Op ( b ∗ a ) = b ( T ) π ( T ) Op ( a ) π ( − T ) d T . G 0 • Decomposition of the Dirac distribution δ : for all m ∈ N , there exist finite families { u j } j ∈ J ∈ U (( g 0 ) C ) and { f j } j ∈ J in C m 0 ( O ) such that for every a ∈ S ′ ( O ) � A ∗ a = ξ 0 ( a ∗ u j ) ∗ f j , j ∈ J where A ∗ ξ 0 ( a )( X ) = a ( Ad ∗ ( − X ) ξ 0 ). Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 11 / 16

Classical applications • G = H 2 n +1 the Heisenberg group • Op : S ′ ( R 2 n ) ∼ → L ( S ( R n ) , S ′ ( R n )) the classical pseudo-diff. Weyl calculus Not. S 0 0 , 0 ( R 2 n ) := { a ∈ C ∞ ( R 2 n ) | ( ∀ α ∈ N 2 n ) ∂ α a ∈ L ∞ ( R 2 n ) } on-Vaillancourt L 2 -boundedness th.) Corollary 1 (Calder´ a ∈ S 0 0 , 0 ( R 2 n ) = ⇒ Op ( a ) ∈ B ( L 2 ( R n )) . Corollary 2 (Beals criterion) If T : S ( R n ) → S ′ ( R n ) cont. lin. and for all r ≥ 1, j = 1 , . . . , r , X j ∈ { Q 1 , . . . , Q n , P 1 , . . . , P n } we have [ X 1 , . . . [ X r , T ] · · · ] ∈ B ( L 2 ( R n )), then there is a ∈ S 0 0 , 0 ( R 2 n ) such that T = Op ( a ). Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 12 / 16

Non-classical examples, pag. 1 The main theorem applies to a wide variety of situations, quite different from the Heisenberg groups: First piece of evidence (L. Corwin, 1983) Every simply connected nilpotent Lie group embeds into a nilpotent Lie group whose generic coadjoint orbits are flat. ⇒ There exist such groups of arbitrarily high nilpotency index. Second piece of evidence (D. Burde, 2006) There exists a nilpotent Lie group with the properties: ◮ the generic coadjoint orbits are flat; ◮ the Lie algebra has only nilpotent derivations. ⇒ Such a group need not be stratified. Daniel Beltit ¸˘ a (IMAR) Symbolic calculus Bia� lowie˙ za, June 27, 2012 13 / 16