The hypoelliptic Laplacian on X = G/K 27 ^ (T XN ),f , f · · Proposition 2.4.2. The connection ^ (T XN ),f , f · is flat. Moreover, . (2.4.11) = (T XN ),f -1 Proof. The first part of the proposition follows from the flatness of the con- nections T XN,f and T XN,f over T X N . Also if e T X, f N , one verifies easily that c (e) c (f ) -1 = -c (e) c (f ) . By (2.4.12), we get c (ad (·)) -1 = -c (ad (·)) . (2.4.13) (2.4.12) Equation (2.4.11) follows from (2.4.1), (2.4.5) and (2.4.13). The proof of our proposition is completed. Remark 2.4.3. Note that in general, the connection (T XN ),f , f does not preserve the total degree of the form. Let K be the universal cover of K. Let S p be the spinors associated with the Euclidean space p. Then c (p) acts naturally on S p . p p p If m is even, then S p is Z 2 -graded, it splits as S p = S + S - , and S ± is m/2-1 m/2 of dimension 2 . Then i is just the involution defining the grading of S p . Also, p p · ^