The hypoelliptic Laplacian on X = G/K 33 When acting on · (p ) L 2 (p), the operator in (2.8.6) is self-adjoint. By (1.5.4), d ,d p p = N S · (p ) + N (p ) . · (2.8.7) By (1.6.10), we can rewrite (2.8.7) in the form · 1 p 1 2 2 - p + |Y p | - m + N (p ) . (2.8.8) (D + E p ) = 2 2 The above operators are K-invariant. Now we consider the vector space k. It is here crucial to consider again the Clifford algebras associated with (k, B| k ). Let e m+1 , . . . , e m+n be an or- thonormal basis of k. We define the operators D k , E k by the formulas m+n D k = i=m+1 c (e ) i e i , E k = c Y k . (2.8.9) Let D k , E k be the operators defined in (1.6.4), which are associated with the Euclidean vector space (k, -B| k ). Comparing the second line of (1.6.11) with (2.8.9), we get D k = D k , E k = -E k . (2.8.10) By (1.6.5) and (2.8.10), we obtain 1 k k k k