At the moment both government deficits and corporate savings are high. It is non-obvious but there is a close relationship between these quantities:
Let corporate profits, P = corporate savings, S + dividends, d
These profits are generated by paying wages to the labour force, W, using the capital stock owned by the corporate sector, K, and producing the economic output of the economy, Y i.e. P = Y - W
Households income is wages, W + dividends, d, less taxes, T, and is spent entirely on consumption, C (i.e. we assume all saving is done by the corporate sector).
The government raises taxes, T and buys government purchases, G from the corporate sector, running a deficit, D = G - T.
The corporate sector can also 'buy' some output from themselves to augment the capital stock, i.e. investment, I is used to increase K.
These relationships can be combined:
P = S + d = Y - W = C + G + I - W = W + d - T + G + I - W = d + D + I
i.e. S = D + I
Therefore, corporate savings is, under some assumptions, equal to government deficits + corporate investment. So when we see corporate savings as the mirror image of government deficits then all we're really seeing is constant (and very low) corporate investment.
This is just accounting, so there's no consideration of incentives or causality here. A freshwater/conservative reading of the current situation might be something like: we take corporate savings as given and governments choose the level of deficit that they run. Therefore government deficits cause the lack of corporate investment by providing an alternative home for these savings.
My reading of the situation is more like, governments choose the level of their spending, G, and they choose the tax rate - but the level of economic activity ultimately determines the tax revenue, T and so deficit, D is endogenous. Expectations of demand determine corporate investment I. Therefore low demand expectations means low investment and is strongly suggestive of low tax take and so high deficits. Corporate savings is still the sum of D and I, but these are driven in opposite directions by demand expectations. If larger government spending now was such that demand expectations were boosted to a level that raised investment, then any inertia in corporate savings would require that the deficit was actually reduced by the increased government spending. Chris Dillow has a post on this too: I don't completely agree with his take on Labour's fiscal policies during the 00's since, given global imbalances it must have been possible to realise that government revenues from the financial sector were not sustainable. If this income had been spent on capital investment rather than revenue spending then his argument is fine though.
Of course it could be that there is no inertia in corporate savings, and so the increase in investment could be accompanied by an increase in corporate savings. This could be achieved by a reduction in dividends - probably reducing tax take and increasing the deficit - and so a reduction in private consumption i.e. higher government consumption merely crowds out private consumption. However, this implies that an economy with unemployed resources chooses to meet an increase in demand from the government by reducing private demand. Quite possible that unemployed resources are used to meet the increase in demand which would allow investment to increase without corporate savings increasing, requiring that the deficit falls.
Again, if we want the investment, required to mitigate resource constraints and climate change, to actually be made, we need to ensure demand is maintained.